An ultimate Sobolev dimension-free imbedding

An ultimate Sobolev dimension-free imbedding Alberto FIORENZA Universitá di Napoli "Federico II" Memorial seminar dedicated to Prof. RNDr. Miroslav Krbec, DrSc. Praha, November 8, 212 () 1 / 25

PRAHA 1995 1996 1998 2 21 212 NAPOLI 1994 1997 23 26 21 211 () 2 / 25

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Sobolev inequality for gradients n 2 Q n = (, 1) n unit cube in R n classical: 1 < p < n p(n 1) u np L n p n p u L p u W 1,p () 4 / 25

Sobolev inequality for gradients n 2 Q n = (, 1) n unit cube in R n classical: 1 < p < n p(n 1) u np L n p n p u L p u W 1,p dimension-free: p > 1 u Xp c p u L p u W 1,p () 4 / 25

Sobolev inequality for gradients n 2 Q n = (, 1) n unit cube in R n classical: 1 < p < n p(n 1) u np L n p n p u L p u W 1,p dimension-free: p > 1 u Xp(,1) = u Xp c p u L p u W 1,p () 5 / 25

Spaces X p such that u Xp c p u L p u W 1,p Poincaré estimate Logarithmic Sobolev inequalities Small Lebesgue inequality Poincaré estimate: u L p p u L p u W 1,p () 6 / 25

Spaces X p such that u Xp c p u L p u W 1,p Poincaré estimate Logarithmic Sobolev inequalities Small Lebesgue inequality Poincaré estimate: special case of u L p p u L p u W 1,p u L p p Ω 1 n u L p u W 1,p (Ω) () 6 / 25

Spaces X p such that u Xp c p u L p u W 1,p Poincaré estimate Logarithmic Sobolev inequalities Small Lebesgue inequality Poincaré estimate: special case of u L p p u L p u W 1,p u L p p Ω 1 n u L p u W 1,p (Ω) The no gain of summability is a first answer! () 6 / 25

Spaces X p such that u Xp c p u L p u W 1,p Poincaré estimate Logarithmic Sobolev inequalities Let s go back to the Sobolev inequality: u L np n p c p,n u L p u W 1,p Small Lebesgue inequality () 7 / 25

Spaces X p such that u Xp c p u L p u W 1,p Poincaré estimate Logarithmic Sobolev inequalities Let s go back to the Sobolev inequality: Small Lebesgue inequality Improvement: u L np n p c p,n u L p u W 1,p where u L np n p,p = u L np n p [ 1 u np c L n p,p p,n u L p u W 1,p (t n p np u (t) ) p dt t ] 1 p, u (t) = 1 t t u (s)ds () 7 / 25

Spaces X p such that u Xp c p u L p u W 1,p Poincaré estimate Logarithmic Sobolev inequalities Small Lebesgue inequality u L np n p,p c p,n u L p u W 1,p O Neil Edmunds, Kerman, Pick Duke Math. J. 1963 J. Funct. Anal. 2 J. Peetre Malý, Pick Ann. Inst. Fourier 1966 Proc. A.M.S. 22 Pick Sobolev Ineq. in Math., Springer 29 () 8 / 25

L np n p,p X p L p L p!!, p! Ln-p! np! X p! () 9 / 25

Spaces X p such that u Xp c p u L p u W 1,p Poincaré estimate Logarithmic Sobolev inequalities Small Lebesgue inequality ( ) 1 1 ( (1 + log t ) p 2 u (t) p dt p c p (true, more generally, for u W 1,p (R n )) Q n u(x) p dx Triebel Banach Center Publ. 211 Martin, Milman J. Funct. Anal. 29, Adv. in Math. 21 ) 1 ) 1 p +c p u(x) ( Q p p dx n () 1 / 25

Spaces X p such that u Xp c p u L p u W 1,p Poincaré estimate Logarithmic Sobolev inequalities Small Lebesgue inequality ( ) 1 1 ( (1 + log t ) p 2 u (t) p dt p c p Q n u(x) p dx ) 1 ) 1 p +c p u(x) ( Q p p dx n u {λ Lp(log L) p = inf > : 2 u Lp(log L) p 2 c p u L p u W 1,p Q n ( u(x) λ ) p ( log p 2 e + u(x) ) } dx 1 λ () 11 / 25

Spaces X p such that u Xp c p u L p u W 1,p Poincaré estimate Logarithmic Sobolev inequalities Small Lebesgue inequality u {λ Lp(log L) p = inf > : 2 Krbec, Schmeisser J. Math. Anal. Appl. 212 Krbec, Schmeisser Rev. Math. Complut. 212 u Lp(log L) p 2 c p u L p u W 1,p Q n ( u(x) λ ) p ( log p 2 e + u(x) ) } dx 1 λ () 12 / 25

Spaces X p such that u Xp c p u L p u W 1,p Poincaré estimate Logarithmic Sobolev inequality: Logarithmic Sobolev inequalities u Lp(log L) p 2 c p u L p u W 1,p Small Lebesgue inequality () 13 / 25

Spaces X p such that u Xp c p u L p u W 1,p Poincaré estimate Logarithmic Sobolev inequality: Logarithmic Sobolev inequalities u Lp(log L) p 2 c p u L p u W 1,p Improvement (F., Krbec, Schmeisser 212): Small Lebesgue inequality where u Lp(log L) p c p 2 (p, p u L 2 p (p, u L 2 c p u L p u W 1,p is a particular small Lebesgue space () 14 / 25

Basics on grand Lebesgue spaces Ω R n bounded domain f : Ω R n Df (x) : R n R n J(x, f ) = det Df (x) x Ω () 15 / 25

Basics on grand Lebesgue spaces Ω R n bounded domain f : Ω R n Df (x) : R n R n J(x, f ) = det Df (x) x Ω Hölder inequality: f W 1,n (Ω, R n ) J(x, f ) L 1 (Ω) () 15 / 25

Basics on grand Lebesgue spaces Ω R n bounded domain f : Ω R n Df (x) : R n R n J(x, f ) = det Df (x) x Ω Hölder inequality: f W 1,n (Ω, R n ) J(x, f ) L 1 (Ω) Müller, Bull. A.M.S. 1989 : f W 1,n (Ω, R n ), J a.e. J(x, f ) L 1 log L loc (Ω) () 15 / 25

Basics on grand Lebesgue spaces Ω R n bounded domain f : Ω R n Df (x) : R n R n J(x, f ) = det Df (x) x Ω Hölder inequality: f W 1,n (Ω, R n ) J(x, f ) L 1 (Ω) Müller, Bull. A.M.S. 1989 : f W 1,n (Ω, R n ), J a.e. J(x, f ) L 1 log L loc (Ω) Iwaniec, Sbordone, Arch. Rat. Mech. Anal. 1992 : Df L n) (Ω), J a.e. J(x, f ) L 1 loc (Ω) () 15 / 25

Basics on grand Lebesgue spaces Ω R n bounded domain f : Ω R n Df (x) : R n R n J(x, f ) = det Df (x) x Ω Hölder inequality: f W 1,n (Ω, R n ) J(x, f ) L 1 (Ω) Müller, Bull. A.M.S. 1989 : f W 1,n (Ω, R n ), J a.e. J(x, f ) L 1 log L loc (Ω) Iwaniec, Sbordone, Arch. Rat. Mech. Anal. 1992 : Df L n) (Ω), J a.e. J(x, f ) L 1 loc (Ω) ( u L n) (Ω) u L n) (Ω) = sup ɛ <ɛ<n 1 Ω ) 1 u(x) n ɛ n ɛ dx < [look at references in Capone, F., J. Funct. Spaces Appl. 25 for the state-of-art until 24] () 15 / 25

Basics on grand Lebesgue spaces Ω R n bounded domain 1 < p < θ > ( u L p),θ (Ω) u L p) (Ω) = sup <ɛ<p 1 ɛ θ Ω ) 1 u(x) p ɛ p ɛ dx L p),θ (Ω) is a rearrangement-invariant Banach Function Space (Bennett, Sharpley 1988) Iwaniec, Koskela, Onninen Invent. Math. 21 L p (log L) θ Lp),θ L Φ < () 16 / 25

Basics on small Lebesgue spaces Ω R n bounded domain 1 < p < θ > Definition The small Lebesgue space L (p,θ (Ω) is the associate space of the grand Lebesgue space L p),θ (Ω) () 17 / 25

Basics on small Lebesgue spaces Ω R n bounded domain 1 < p < θ > Definition The small Lebesgue space L (p,θ (Ω) is the associate space of the grand Lebesgue space L p),θ (Ω) } u L (p,θ (Ω) { = sup u(x)v(x) dx : v L p),θ (Ω), v L p),θ (Ω) 1 Ω u(x)v(x)dx u L (p,θ (Ω) v L p),θ (Ω) Ω L (p,θ (Ω) is a rearrangement-invariant Banach Function Space () 17 / 25

Basics on small Lebesgue spaces Ω R n bounded domain 1 < p < θ > () 18 / 25

Basics on small Lebesgue spaces Ω R n bounded domain 1 < p < θ > F. F., Rakotoson Collect. Math. 2 Math. Ann. 23 { ( ) 1/(p ɛ) } u L (p,θ (Ω) inf u= P k u k ɛ) inf <ɛ<p 1 ɛ θ/(p u k (p ɛ) dx k Ω () 18 / 25

Basics on small Lebesgue spaces Ω R n bounded domain 1 < p < θ > F. F., Rakotoson Collect. Math. 2 Math. Ann. 23 { ( ) 1/(p ɛ) } u L (p,θ (Ω) inf u= P k u k ɛ) inf <ɛ<p 1 ɛ θ/(p u k (p ɛ) dx k F., Karadzhov Di Fratta, F. Z. Anal. Anwen. 24 Nonlinear Anal. 29 u L (p,θ (Ω) Ω ( ( )) θ t p 1 log 1 ( t ) 1 p u (s) p dt ds Ω t Ω () 18 / 25

Basics on small Lebesgue spaces Capone, F. J. Funct. Spaces Appl. 25 Capone, F., Krbec J. Ineq. Appl. 26 Cobos, Kühn preprint 212 L p (log L) βθ/(p 1) (Ω) L (p,θ (Ω) L p (log L) θ/(p 1) (Ω) β>1 In particular, when θ = p /2, i.e. L (p,p /2 (Ω) L p (log L) p/2 (Ω) u L p (log L) p/2 (Ω) u L (p,p /2 (Ω) () 19 / 25

Grand/small Sobolev spaces Fusco, Lions, Sbordone, Proc. A.M.S. 1996 u W 1,1 (Ω), u Ln),θ (Ω) u EXP(L n/(n 1+θ) )(Ω) () 2 / 25

Grand/small Sobolev spaces Fusco, Lions, Sbordone, Proc. A.M.S. 1996 u W 1,1 (Ω), u Ln),θ (Ω) u EXP(L n/(n 1+θ) )(Ω) F., Rakotoson, Math. Ann. 23 u W 1,1 (Ω), u L (n (Ω) osc B(x,t) u c N Ω 1/N u χ B(x,t) L (n (Ω) () 2 / 25

Grand/small Sobolev spaces Fusco, Lions, Sbordone, Proc. A.M.S. 1996 u W 1,1 (Ω), u Ln),θ (Ω) u EXP(L n/(n 1+θ) )(Ω) F., Rakotoson, Math. Ann. 23 u W 1,1 (Ω), u L (n (Ω) osc B(x,t) u c N Ω 1/N u χ B(x,t) L (n (Ω) F., Rakotoson, Calc. Var. 25 (case k = 1) Cobos, Kühn, preprint 212 (case k positive integer) Ω R n connected, bounded, open, smooth, Ω = 1, 1 < p < n/k W k,p (Ω) L p ) are compact, where p = np/(n kp) W k,(p (Ω) L p () 2 / 25

F., Krbec, Schmeisser, the main idea Alvino, Boll. U.M.I. 1977 u np/(n p),p c p n 1/2 u p u W 1,p () 21 / 25

F., Krbec, Schmeisser, the main idea Alvino, Boll. U.M.I. 1977 u np/(n p),p c p n 1/2 u p u W 1,p n 1/2 u np/(n p),p c p u p () 21 / 25

F., Krbec, Schmeisser, the main idea Alvino, Boll. U.M.I. 1977 u np/(n p),p c p n 1/2 u p u W 1,p n 1/2 u np/(n p),p c p u p inf n n1/2 u np/(n p),p c p u p () 21 / 25

F., Krbec, Schmeisser, the main idea Alvino, Boll. U.M.I. 1977 u np/(n p),p c p n 1/2 u p u W 1,p n 1/2 u np/(n p),p c p u p inf n n1/2 u np/(n p),p c p u p (change n 1/ɛ) inf ɛ ɛ 1/2 u (p ɛ),p c p u p () 21 / 25

F., Krbec, Schmeisser, the main idea Alvino, Boll. U.M.I. 1977 u np/(n p),p c p n 1/2 u p u W 1,p n 1/2 u np/(n p),p c p u p inf n n1/2 u np/(n p),p c p u p (change n 1/ɛ) inf ɛ ɛ 1/2 u (p ɛ),p c p u p (setting θ = p /2) ɛ 1/2 = ɛ θ/p ɛ θ/(p ɛ) () 21 / 25

F., Krbec, Schmeisser, the main idea Alvino, Boll. U.M.I. 1977 u np/(n p),p c p n 1/2 u p u W 1,p n 1/2 u np/(n p),p c p u p inf n n1/2 u np/(n p),p c p u p (change n 1/ɛ) inf ɛ ɛ 1/2 u (p ɛ),p c p u p (setting θ = p /2) (conclusion) ɛ 1/2 = ɛ θ/p ɛ θ/(p ɛ) u (p,p /2 c p u p () 21 / 25

F., Krbec, Schmeisser, the main idea Alvino, Boll. U.M.I. 1977 u np/(n p),p c p n 1/2 u p u W 1,p u Y c p n 1/2 u np/(n p),p c p u p inf n n1/2 u np/(n p),p c p u p (change n 1/ɛ) (setting θ = p /2) (conclusion) optimality : L (p,p /2 Y inf ɛ ɛ 1/2 u (p ɛ),p c p u p ɛ 1/2 = ɛ θ/p ɛ θ/(p ɛ) u Y c p u (p,p /2 c p u p () 22 / 25

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