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(1) (pl)[yFy xFx]Fyy [{p l)Fy xFxy] 2 + xFxxFyy = 0 for F nonconstant and satisfying other conditions on a suitable domain of R. We assume throughout that 1 < p < oo and write LP for the real Lebesgue space LP(0,1). The unconditional constant Kp(e) of a sequence e = (ei, e2,...) in LP is the least K e [ l , +oo] with the property that if n is a positive integer and o i , . . . ,a n are real numbers such that || Y*k=iafcefc||p = 1, then I n I \J2 kk^k\ l*-l lip for all choices of signs €& G {—1,1}. The sequence e is a basis of L if, for every ƒ G LP, there is a unique sequence o such that || ƒ — ]Cfc=i k^k\\p —• 0 as n —• oo. A sequence d = (di ,^, . . . ) in LP is a martingale difference sequence if dn+i is orthogonal to <p(dii...idn) for all bounded continuous functions <p: R • Rand all n > l . The Haar system h = (fei,/i2,...) is both a basis of L (Schauder [11]) and a martingale difference sequence: /in+i is supported by a set on which p(/ii,. . . ,/in) is constant. By an inequality of Paley (see [10 and 6]), its unconditional constant Kp(h) is finite. More generally [2], supd Kp(d) is finite where the supremum is taken over all martingale difference sequences d in L. In fact, if d is a martingale difference sequence and e is a basis of L, then (2) Kp{d)<Kp(h)<Kp(e).

A nonlinear partial differential equation and the unconditional constant of the Haar system in 𝐿^{𝑝}