Poduct Rule and Chain Rule Estimates fo Hajlasz Gadients on Doubling Metic Measue Saces A Eduado Gatto and Calos Segovia Fenández Ail 9, 2004 Abstact In this ae we intoduced Maximal Functions Nf, x) of AP Caldeón in the context of doubling metic measue saces X, d, µ) It is shown that these maximal functions ae equivalent to the Hajlasz gadients Using these maximal functions we ove L s nom estimates fo the Poduct Rule and the Chain Rule fo functions on X, d, µ) Intoduction We shall say that X, d, µ) is a metic measue sace if X, d) is a metic sace and µ is a Boel measue on X Let B x) denote the ball of cente x and adius > 0 If µ satisfies the condition µb 2 x)) C D µb x)) with a constant C D indeendent of x and, then X, d, µ) will be called a doubling metic measue sace Following Hajlasz we shall say that a measuable function f has a gadient g in L X, d, µ) if fx) fy) dx, y)gx) + gy)) ) holds fo all x and y X with g L It can be shown that fo > thee exists a unique g = f in L satisfying ) and such that f = inf g g, whee the infimum is taken ove all g satisfying ) We will call f the Hajlasz gadient of f in L Let f be a measuable function and u, we define the Caldeón Maximal Function N u f) by /u N u f, x) = su fy) fx) dµy)) u >0 µb x))
We show in Theoem the elationshi between f and N u f) In Theoem 2 we ove the main estimate fo N u F ) that is needed to obtain the ules We ove in Theoem 3 the Poduct Rule Estimate and in Theoem 4 the Chain Rule Estimate 2 Theoems and Poofs Theoem Let X, d, µ) be a doubling metic measue sace, f a measuable function, < <, and u < a If f has the Hajlasz gadient in L, then thee is a constant C u, ) such that N u f) C u, ) f b If N u f) is in L, the f has the Hajlasz gadient in L and thee is a constant C 2 u, ) such that f C 2 u, ) N u f) Poof We shall ove fist at a) µb x) Since f satisfies ), we have fy) fx) u dµy)) /u [ f x) + f y) ) /u u dµy) µb x)) f [ x) + f y) ) /u u dµy) µb x)) Dividing both sides by and taking suemum we get N u f, x) f x) + M u f )x) whee M u f )x) = su B x) > u, M u f ) Cu, ) f, thus We shall ove now at b) µ) [ f y) u dµy) ) /u Since N u f, x) C u, ) f Let x and y in X and = dx, y) We have /u fx) fy) fx) fz) dµz)) u + µb x) 2
dx, y) µb x)) dx, y) µb x) fz) fy) u dµz)) /u fx) fz) u dµz)) /u + [ /u ) /u µb2 y) fz) fy) u dµz) µb x)) µb 2 y)) B 2y) [ dx, y) C 2/u D N uf, x) + C 2/u D N uf, y) 2)) Theefoe, fom the definition of f it follows that f C 2/u D N uf) Theoem 2 Let u < s and s = + q, <, < q Then su >0 µb x)) fy) fx) u hy) u dµy)) /u C N u f) h q, whee C is a constant indeendent of f and h Poof We use inequality 2) to get µb x)) µb x)) CD 2 µb x) fy) fx) u hy) u dµy)) /u s d u x, y)c 2 D [N u f, x) + N u f, y) u hy) u dµy)) /u N u f, x)c 2/u D C 2/u D [N u f, x) + N u f, y) u hy) u dµy)) /u µb x)) µb x)) hy) u dµy)) /u + N u u f, y) hy) u dµy)) /u C 2/u D N uf, x)m u h)x) + C 2/u D M un u f) h)x) 3
Now, taking the suemum on the left hand side, then the L s nom of both sides, and using that u < s q we obtain /u su fy) fx) u hy) dµy)) u >0 µb x)) s C 2/u D [ N uf) M u h) s + M u N u f) h) s CD [ N 2 u f) M u h) q + N u f) h s 2C 2 D N u f) h q Theoem 3 Poduct Rule Estimate) Let <, 2 <, < q, q 2, < s, < u < s and s = + 2 q Then thee is a constant C indeendent of f and g such that 2 N u fg) s C [ N u f) g q + N u g) 2 f q2 Poof We wite N u fg)x) = su su su su µb x)) µb x)) µb x)) µb x)) fy)gy) fx)gx) u dµy) fy) fx) u gy) u dµy)) /u + gy) gx) u fx) u dµy)) /u + q = ) /u fy) fx) u gy) u dµy)) /u + N u g, x)fx) We comute now the L s nom of both sides, and use Theoem 2 and Hölde s inequality to estimate the tems on the ight hand side esectively we obtain N u fg) s C [ N u f) g q + N u f) 2 g q2 Theoem 4 Chain Rule Estimate) Let F C C), Hz) = q, u < s Then su w < z with C indeendent of F and g F w), s = + q N u F g) s C H g N u g) q, < s, <, < 4
Poof Obseve that su F λz + λ)z 2 ) Hz )+Hz 2 ) fo any z, z 2 0λ Then alying the Mean Value Theoem we have, F gy)) F gx)) su F λgy) + λ)gx)) gy) gx) 0λ [Hgy)) + Hgx)) gy) gx) Theefoe N u F g, x) = su >0 µb x)) su µb x) ) /u F gy)) F gx) u dµy) gy) gx) u H u gy))dµy)) /u + Hgx))N u g, x) We comute now the L s nom of both sides, and using Theoem 2 and Hölde s inequality to estimate the tems on the ight hand side esectively, we obtain N u F g) s C H g N u g) q which concludes the oof of Theoem 4 Note: The exlicit fomula fo H in Theoem 4 was suggested by Michael Chist Pesonal Communication Refeences [ Caldeon s, A P Estimates fo singula integal oeatos in tems of maximal functions, Studia Mathematics, T XLIV 972) [2 Gatto, A E, Poduct Rule and Chain Rule Estimates fo Factional Deivatives on Saces that satisfy the doubling condition Jounal of Functional Analysis 88, 27-37 2002) [3 Hajlasz, P, Sobolev saces on an abitay metic sace Potential Anal 5996) 403-45 [4 Heinonen, J, Lectues on Analysis on Metic Saces Singe-Velag, 200, New Yok 5
A Eduado Gatto Deatment of Mathematics DePaul Univesity Chicago, IL 6064 USA e-mail: aegatto@condodeauledu Calos Segovia Fenández Instituto Agentino de Matemática IAM-CONICET) Saveda 5,Buenos Aies Agentina e-mail: segovia@iambaedua 6