DIANE P. WOOD, Circuit Judge.

Almost the only thing left of this case, which originally involved a dispute about trademark rights, is a peripheral fight over sanctions and attorneys' fees. The present appeals are from an order granting $100,000 in sanctions against three plaintiffs and holding the plaintiffs' lead counsel jointly liable for 25% of that amount. The plaintiff-appellants have also made a perfunctory argument that the district court should not have granted summary judgment to the defendant, but those arguments, which consume exactly eight lines of a 45-page brief, are undeveloped and therefore waived. Colburn v. Trustees of Indiana Univer., 973 F.2d 581, 593 (7th Cir. 1992). The defendants have cross-appealed, seeking to have the plaintiffs' lead counsel (and his law firm) held jointly liable for the full amount of the sanctions, to have the amount of sanctions increased to $800,000, and to subject the plaintiffs' local counsel to some amount of sanctions as well. We find no abuse of discretion in the district court's decision to grant sanctions, and we therefore affirm its decision on the appeal. Because the judge did not offer any explanation for the amount he chose, however, we remand for his further consideration of that point.

I.

Hans Maurer AG ("Maurer") is a Swiss company that holds the patent and trademark rights to the CLOSOMAT brand of toilet designed for "paperless" use. Maurer licensed its invention to one of the plaintiffs, the Finance Investment Company ("FIC"). The other two plaintiffs in this lawsuit — Closomat US ("Closomat") and the Joseph Muller Corporation Zurich ("JMCZ") — were authorized distributors of the CLOSOMAT toilet (not, it may be worth noting, sublicensees). Joseph Muller controlled all three of the plaintiff companies, but Muller himself is not a party to this lawsuit. For the sake of simplicity, we will refer to the plaintiffs collectively as the Muller companies. Evidently, Maurer's effort to develop and sell a "paperless" toilet was not purely quixotic. The defendant companies, Geberit Manufacturing Inc. and its parent Geberit AG (collectively, "Geberit," which is a large Swiss manufacturer of plumbing equipment), apparently hold the rights to a competing "paperless" toilet of their own. In fairness to the concept of "paperless" toilets — which some skeptics have likened to the elusive "paperless office," see David Harrison, The Observer, Oct. 5, 1997 — we note the existence of a niche market for the physically disabled, see Eric Rich, State of the Art House Accommodates Age-Related Disabilities, Hartford Courant, June 21, 1998, and the fact that there is apparently a growing market outside the United States for these devices. See Harry Bruce, Ottawa Citizen, Jan. 21, 1989 (stating that the $5,000 toilets are "getting popular" in Europe).

The trouble giving rise to this lawsuit began with an article that appeared in the March 1989 issue of Sala Bañ o, a bathroom fixtures trade magazine distributed mostly in Europe but with a small circulation in the United States. The article pictured a Geberit toilet but repeatedly and incorrectly referred to it as a "Geberit, series `Closomat'" device. When Muller learned of the article's misidentification of the Geberit product, he complained to the editors of Sala Bañ o, who apologized in a letter. Unmollified, Muller caused FIC, Closomat, and JMCZ to sue Geberit and the publishers of Sala Bañ o, Faenza Editrice Iberica S.L. ("Faenza"), in January 1992 in the U.S. District Court for the Southern District of Florida. The Florida complaint alleged that Geberit (with Faenza's negligent assistance) had violated various provisions of the Lanham Act. The Muller companies sought injunctive relief and damages of "no less than" $12,000,000 plus treble damages and attorneys' fees. Lead counsel Andrew Goodman signed these pleadings for the plaintiffs.

Geberit officials filed uncontested affidavits at summary judgment swearing that they had nothing to do with the Sala Bañ o article. In December 1992, the Florida court entered summary judgment in favor of Geberit. It found that the plaintiffs' failure to controvert the Geberit affidavits left nothing in the record suggesting that the company had committed a tort in the state of Florida, and thus it concluded that the court lacked in personam jurisdiction over Geberit. The plaintiffs filed a motion for reconsideration under Fed.R.Civ.P. 60(b); Geberit responded with a motion for sanctions and entry of judgment under Fed.R.Civ.P. 54(b). In February 1993, the court denied the plaintiffs' Rule 60(b) motion, granted Geberit's Rule 54(b) motion, and denied the motion for sanctions. The plaintiffs appealed the Rule 54(b) final judgment to the Eleventh Circuit, which affirmed per curiam without opinion. Finance Investment Co. v. Geberit, 11 F.3d 167 (11th Cir. 1993). The Florida district court did not enter a judgment on the plaintiffs' claims against Faenza.

In April 1993, a few months after the Florida court granted summary judgment, the same Muller companies filed a substantially identical complaint against Geberit in the U.S. District Court for the Northern District of Indiana. (One of the Geberit companies maintains its principal place of business in Indiana.) Again, lead counsel Goodman signed the complaint, though he was joined this time by local counsel Frederick Eichhorn. In relevant part, the complaint alleged that Geberit knowingly paid for, drafted, or otherwise assisted Faenza with the publication of the Sala Bañ o article in violation of the Lanham Act. In February 1994, the remnants of the Florida lawsuit were transferred on the Muller companies' motion under 28 U.S.C. § 1404(a) to the Indiana court, and the two actions were consolidated in July 1994. In February 1995, Geberit moved for summary judgment on all of the plaintiffs' claims; the district court granted that motion in July 1995 and ordered the Muller companies and their counsel to show cause why they should not be required to pay Geberit's reasonable attorneys' fees. In August 1995, the Muller companies filed a notice of appeal from the grant of summary judgment.

The notice of appeal did not stop events related to the fee petition from proceeding in the district court. The following week, Geberit filed a motion for $800,000 in attorneys' fees and expenses. Plaintiffs' counsel responded to both Geberit's motion and the district court's order to show cause. So things remained for nearly two years — though Geberit and the Muller companies continued to fire back and forth a steady stream of memoranda on the outstanding sanctions issue and various other matters. In May 1997, this court suspended the pending appeal from the July 1995 summary judgment and ordered counsel to file a report about the status of proceedings at the district court level. A week later, the district court ruled on the outstanding sanctions issue, as well as all other pending motions, and ordered the Muller companies to pay $100,000 in "monetary sanctions toward their attorney's fees" to Geberit. The order made lead attorney Goodman jointly liable for 25% of this amount; local counsel Eichhorn escaped with an admonishment. The district court also dismissed the Muller companies' left-over claim against Faenza from the Florida litigation. The court then entered the May 1997 order pursuant to Fed.R.Civ.P. 58 and 79(a). The Muller companies, Goodman, and Eichhorn filed a timely appeal challenging the grant of sanctions; Geberit cross-appealed.

II.

The district court's May 1997 order granting sanctions (directed toward attorneys' fees) discussed three alternative grounds for its decision: Fed.R.Civ.P. 11, 28 U.S.C. § 1927, and 15 U.S.C. § 1117. We review a decision to grant sanctions under any of these grounds for an abuse of discretion. See Anderson v. County of Montgomery, 111 F.3d 494, 501 (7th Cir. 1997) (Rule 11); Fox Valley Construction Workers' Fringe Benefit Funds v. Pride of the Fox Masonry Expert Restorations, 140 F.3d 661, 666 (7th Cir. 1998) (sec. 1927); BASF Corp. v. Old World Trading Co., Inc., 41 F.3d 1081, 1092 (7th Cir. 1994) (sec. 1117). We review the factual findings underlying the district court's imposition of sanctions for clear error, Melendez v. Illinois Bell Tel. Co., 79 F.3d 661, 671 (7th Cir. 1996), and the choice of sanction for an abuse of discretion. Id. at 670. Finally, we are not bound by the district court's specific reasoning and may affirm a grant of sanctions on any basis supported by the record and the law. In re Volpert, 110 F.3d 494, 500 (7th Cir. 1997). The appellants have their work cut out for them.

Generally speaking, the district court granted sanctions for two reasons. First, it concluded that the Muller companies were not entitled to sue under the Lanham Act as a matter of law; second, it determined that they filed suit knowing that their claims lacked substantive merit. Regarding the first issue, the court found that: (1) the Muller companies' Indiana complaint falsely asserted that Closomat was sublicensed by FIC; (2) all three plaintiff companies did not have the kind of interest in the trademark that entitled them to sue under sec. 32(1) of the Lanham Act; (3) under the express terms of the license, FIC (and consequently all the other plaintiffs) could not bring suit under sec. 43(a) of the Lanham Act, either; and (4) the plaintiffs' attorney Goodman should have known that his clients did not have a proper Lanham Act claim and that, as litigation proceeded, the suit should have been dismissed. Regarding the lack of substantive merit to the plaintiffs' claims, the court found that: (1) the plaintiffs and their attorneys sued in Indiana despite knowing from the previous Florida litigation that Geberit "had nothing to do with . . . [the] publication or alleged distribution" of the Sala Bañ o article — knowledge that rendered their substantive claims "wholly without foundation"; and (2) none of the information produced by Geberit during discovery suggested that Geberit had ever attempted to trade on the CLOSOMAT mark.

The brief on behalf of "plaintiffs-appellants-cross-appellees" on appeal, which was filed on behalf of the Muller companies, Goodman, and Eichhorn, dedicates almost the entirety of its attention to the district court's conclusion that FIC, Closomat, and JMCZ lacked what they call "standing to sue" under §§ 32(1) and 43(a) of the Lanham Act. (They are not referring to Article III standing; they refer instead to the argument that FIC, Closomat, and JMCZ did not satisfy the statutory requirements for a Lanham Act claim.) This is a high-risk strategy, for we have repeatedly held that where the district court has rendered one or more alternative holdings on an issue, the failure to address one of the holdings results in a waiver of any claim of error with respect to the court's decision on that issue. Kauthar SDN BHD v. Sternberg, 149 F.3d 659, 668 (7th Cir. 1998); Williams v. Leach, 938 F.2d 769, 772 (7th Cir. 1991); Landstrom v. Illinois Dep't of Children Family Servs., 892 F.2d 670, 678 (7th Cir. 1990). Not until their reply brief do the appellants address the district court's finding that the Florida lawsuit put the Muller companies on notice that Geberit "had nothing to do with . . . [the] publication or alleged distribution" of the Sala Bañ o article. However, arguments raised in the reply brief are also waived. James v. Sheahan, 137 F.3d 1003, 1008 (7th Cir. 1998). Thus, we would be justified in affirming the sanctions granted simply because the appellants waived any challenge to the district court's finding that they filed suit despite having knowledge that Geberit had been innocent of wrongdoing. The discussion that follows should therefore be regarded solely as a ruling in the alternative by this court, to forestall any argument that the five appellants somehow avoided waiver.

A. The Muller Companies' Alleged Lanham Act Claims

Maurer licensed the CLOSOMAT mark to FIC; it is undisputed that neither Closomat nor JMCZ were licensees. All three plaintiff companies sued under §§ 32(1) and 43(a) of the Lanham Act. Section 32(1) of the Lanham Act, 15 U.S.C. § 1114(1), creates a cause of action for a trademark "registrant" when a defendant uses a registered trademark without consent. Section 43(a) of the Lanham Act, 15 U.S.C. § 1125(a), creates a cause of action for "any person who believes that he or she is likely to be damaged" by a defendant's false designations of origin and misleading representations of fact that are likely to cause confusion as to origin or sponsorship. The question of who may enforce each provision is the one now before us. Authorities uniformly agree that only the trademark's registrant (or her assignee) may sue under sec. 32(1). See 4 J. Thomas McCarthy, McCarthy on Trademarks and Unfair Competition, §§ 32:3, 32:18 (4th ed. 1998) (citing cases); 3 Jerome Gilson, Trademark Protection Practice sec. 8.16[1][b], at 8-360 (1997) (citing cases and agreeing that this is true even where the licensee holds a "quasi-exclusive" license pursuant to which the registrant-licensor has covenanted not to award competing licenses to others in the region but retains certain rights to the mark). This rule is consistent with 15 U.S.C. § 1127, which defines the term "registrant" to include assignees but not licensees.

Given this rule, mere distributors like Closomat and JMCZ are not authorized enforcers of sec. 32(1). See Gilson, sec. 8.16[1][c] n. 20 (citing cases). The Muller companies concede this and note that they voluntarily withdrew their sec. 32(1) claims on behalf of Closomat and JMCZ prior to the order to show cause. They persist in arguing, however, that as an "exclusive licensee" FIC was entitled to raise a sec. 32(1) claim. In theory, there might be a glimmer of hope for this argument, for "a truly exclusive licensee, one who has the right even to exclude his licensor from using the mark . . . is equated with an assign[ee] since no right to use [the mark] is reserved to the licensor, and the licensee's standing derives from his presumed status as an assignee." Gilson, sec. 8.16[1][b]. Cf. Quabaug Rubber Co. v. Fabiano Shoe Co., 567 F.2d 154, 159 (1st Cir. 1977). Thus, FIC's argument that it had a statutory right to sue as an "exclusive licensee" depends on the language of its license agreement with Maurer. One glance at this document, however, puts to rest any argument that FIC's "exclusive license" was tantamount to an assignment, for the license set forth many duties and rights between the parties that are inconsistent with an assignment, such as geographic limitations on the licensee's territory. Accordingly, the district court did not err in concluding that all of the plaintiffs, including FIC, were not entitled to sue under sec. 32(1).

Although sec. 43(a) confers a right to sue on a broader range of plaintiffs than sec. 32(1), the same text of the license agreement that doomed FIC's sec. 32(1) argument also dooms all three companies' argument that they had a claim under sec. 43(a). Even assuming they met the statutory requirement of being a "person who believes that he or she is likely to be damaged" by a likelihood of confusion, the express terms of the license prohibited any of them from bringing suit in their own capacity. Under sec. 5.8 of the agreement, if FIC had knowledge of an infringement, it was under a duty to notify Maurer. At that point — after notice was given — under sec. 4.5 of the license Maurer was obligated to bring suit. Only if Maurer failed to sue could FIC institute its own action as a plaintiff. FIC failed to give notice to Maurer before suing, and the district court expressly found that no admissible evidence contradicted the assertion that Maurer never gave consent for the plaintiffs to bring suit in their own right. Because the license is the sole source giving the plaintiffs any interest in the CLOSOMAT mark, that same license's refusal to give them the right to sue under these circumstances strips them of the right to raise a sec. 43(a) claim. Once again, the district court did not err in this regard.

B. Substantive Merit of Plaintiffs' Claims

The district court also based its grant of sanctions on its finding that the Muller companies, assisted by their lawyers, brought the Indiana suit knowing that they lacked proof Geberit had done anything wrong. The record supports that finding, and as we mentioned, we are hard pressed to find a challenge to this finding in the briefs on appeal. The Florida litigation ended in December 1992; the plaintiffs brought their Indiana lawsuit in April 1993. It is thus painfully clear that by the time the Muller companies filed the Indiana action, they already knew that, in the course of dismissing their Florida lawsuit against Geberit for lack of in personam jurisdiction, the Florida district court had found that "[n]one of the Plaintiffs' evidence contradict[ed] the Defendants' affidavits which state[d] that they did not pay for or have anything to do with the [Sala Bañ o] article/advertisement." Likewise, the publisher of Sala Bañ o had written to the plaintiffs apologizing for its own error. Nevertheless, the Muller companies refiled the same lawsuit in Indiana, pleading once again "upon information and belief . . . [that Geberit] knowingly" paid for, participated in, or otherwise assisted with the creation of the Sala Bañ o article. What new information had fallen into their hands as a result of pre-filing investigation? The plaintiffs allude to Geberit documents produced during discovery, but those documents were not obtained pre-filing and, in any event, do not help them. For example, one Geberit memo discussing a planned change of name for its "paperless" toilet stated that "[a] requirement for a new name would be . . . separation from the name CLOSOMAT." (Emphasis added.) Far from seeking to trade on the CLOSOMAT mark, the memo proposed that Geberit use the names "Aquasan" or "Clo2000." The district court inspected the documents at summary judgment — as have we — and it properly concluded that the plaintiffs' suspicions about Geberit were "wholly without foundation."

III.

The district court articulated three alternative grounds for its sanctions order: the pre-1993 amendment version of Fed.R.Civ.P. 11; 28 U.S.C. § 1927; and 15 U.S.C. § 1117. Because the Indiana lawsuit was filed in April 1993, which was before the December 1 effective date of the 1993 amendment to Rule 11, the court correctly looked to the pre-1993 text. See In re Generes, 69 F.3d 821, 826-27 (7th Cir. 1995); Cunningham v. Waters Tan Co., 65 F.3d 1351, 1361 (7th Cir. 1995). Under that version, an attorney's signature on a pleading is a representation to the court that "the signer has read the pleading . . . [and that] to the best of the signer's knowledge, information and belief formed after reasonable inquiry it is well grounded in fact and warranted by existing law or good faith argument for the extension, modification, or reversal of existing law. . . ." Fed.R.Civ.P. 11 (1993). The rule also specifies that in the event of a violation "the court . . . shall impose upon the person who signed [the paper], a represented party, or both, an appropriate sanction, which may include an order to pay the other party or parties the amount of the reasonable expenses incurred because of the filing of the pleading, motion, or other paper, including a reasonable attorney's fee." Id. (emphasis added).

For the reasons we have just discussed, lead counsel Goodman's signing of the Indiana complaint violated Rule 11. He had already learned in the Florida litigation that none of the plaintiffs had a colorable claim to standing under §§ 32(1) or 43(a). At oral argument, Goodman told this court that he had "taken his client's word" that Maurer had authorized the lawsuit, and thus that at least FIC could sue under the terms of the license. Apart from revealing a wholly inadequate pre-filing investigation, the truth of this assertion is irrelevant. Even assuming Goodman's client had made such a representation, it was still clear that Closomat and JMCZ had no right whatever under the statute to pursue the litigation, that FIC itself had nothing resembling an assignment, that there was no concrete evidence in Goodman's hands from Maurer, that the language of §§ 4.5 and 5.8 of the License Agreement did not support an independent suit by FIC, and that there was no underlying factual basis for the suit. This is more than enough to justify the Rule 11 sanctions the district court imposed.

The district court also justified its award of sanctions against Goodman under 28 U.S.C. § 1927, which permits a court to award costs, expenses, and attorney fees against a lawyer who "multiplies the proceedings in any case unreasonably and vexatiously." All of the reasons that militate in favor of sanctioning Goodman under Rule 11 exist in spades for the district court's sanction order under sec. 1927. Goodman's decision to file the Indiana suit after the Florida loss was certainly an unreasonable multiplication of proceedings, and it thus fits sec. 1927 to a "T." Furthermore, even if the Florida proceedings had not revealed the infirmity of the plaintiffs' claims against Geberit, Goodman's own review of the documents Geberit provided during discovery in the Indiana case should have clarified the matter and led to a prompt dismissal of the action. See Dahnke v. Teamsters Local 695, 906 F.2d 1192, 1201 n. 6 (7th Cir. 1990). By then, Geberit had submitted an affidavit from the publisher of Sala Bañ o testifying that the misidentification in the article was a "mistake" and that Geberit had not been involved in any way. In addition to the sanctions under sec. 1927, the court invoked its power under 15 U.S.C. § 1117 to make the Muller companies (as opposed to their counsel, Goodman) jointly liable for Geberit's attorneys' fees. That statute allows attorney fee shifting in "exceptional" Lanham Act cases. Although sec. 1117 generally operates as an expense-and-fee shifting statute for prevailing parties against infringers, it may in exceptional circumstances, and when proven by clear and convincing evidence, "provide protection against unfounded suits brought by trademark owners for harassment and the like." Otis Clapp Son, Inc. v. Filmore Vitamin Co., 754 F.2d 738, 746 (7th Cir. 1985); Machinery Corp. of America v. Gullfiber AB, 774 F.2d 467, 471 (Fed. Cir. 1985). In Door Systems Inc. v. Pro-Line Door Systems Inc., 126 F.3d 1028, 1032 (7th Cir. 1997), we recently held that to qualify for fees under sec. 1117(a), a prevailing defendant wrongfully sued needs to show that the lawsuit filed against it can fairly be described as "oppressive."

We have no doubt that the plaintiffs' prosecution of this lawsuit meets that standard. The district court docket sheet is thick with activity, including protracted discovery disputes. The Indiana litigation alone has been ongoing for more than four years. Because of the lawsuit's international dimensions, many documents had to be translated from German to English. The district court had every reason to characterize the Indiana litigation as "oppressive." We therefore affirm the court's judgment to award sanctions in principle.

IV.

The question whether we can affirm the amount the district court chose, along with the allocation between lawyer and client it selected, is more difficult. In total, Geberit requested over $800,000 in sanctions; the district court awarded $100,000. Unfortunately, the court explained the level of the award only with a general reference to the reasoning in its order as a whole. This reduces us to speculation on the court's reasons. It is quite likely that the Muller plaintiffs and Goodman have nothing to complain about. The record includes evidence of the bills incurred by Geberit during this litigation, and that evidence suggests that something well in excess of the $100,000 could have been justified. For example, although some of the bills reflected the Florida phase of the litigation, the record also shows that over the course of one year, counsel for Geberit during the Indiana proceedings charged its client over $315,000. We cannot tell, however, why the court awarded an amount one-eighth the size of the amount Geberit was prepared to support, and less than one-third the fees for the year in question.

This question takes us directly to Geberit's cross-appeal. Geberit argues that the amount of sanctions should be increased to $800,000, that attorney Goodman should be held jointly liable for 100% of this amount, that Goodman's law firm should be added under Rule 11 as a liable party, and that some lesser amount of sanctions should be imposed on local counsel Eichhorn. On the basis of this record, we are not in a position to make an informed judgment on these matters, because we do not know why the district judge discounted the award as it did. Based on the way the parties have conducted this litigation before our court, the judge may have considered Geberit's bill to be grossly inflated. Or the judge may have had other reasons for his choice. These matters lie in the sound discretion of a district judge, and we rarely disturb a district judge's reasoned decision to choose a particular level for sanctions. But as the record now stands, we think it the better part of prudence to remand the cross-appeal to the district court so that it can provide the necessary explanation. At that point, of course, any further review by this court would be only for abuse of discretion.

Before closing, we offer a warning to Geberit. In the May 1997 order granting sanctions, the district court expressly stated that it was not resolving Geberit's motion for costs pursuant to 28 U.S.C. § 1920 and Fed.R.Civ.P. 54(d)(1). In their appellate brief, Geberit reminded this court that the issue of its costs was preserved and that "[i]f necessary, [the] . . . application [for costs] will be made [later]." We do not look forward to another round of extended collateral litigation and appeal. Before Geberit hurries back to the trough for seconds, we remind it of the adage that "pigs get fat, but hogs get slaughtered." In litigating the present appeal, Geberit's lawyer has filed some ridiculous motions, including a "Motion to Correct Brief" in which counsel requested leave to change the position of a quotation mark. Such filings are frivolous, sanctionable in this court under Fed.R.App.P. 38, and unethical if done for the purpose of racking up billable hours. We admonish Geberit and its counsel to remember that any application for costs, if filed, will be subject to the rigors of (new) Rule 11 as well as the district court's inherent authority to sanction. The district court should take great care in examining all of Geberit's claimed costs.

The judgment of the district court is Affirmed in part and the case is Remanded in part for further proceedings consistent with this opinion. Each party shall bear its own costs on appeal.