on interpolation of analytic families of multilinear operators · the classical...

On interpolation of analytic familiesof multilinear operators

Mieczysław Mastyło

Adam Mickiewicz University, and Institute of Mathematics,Polish Academy of Sciences (Poznań branch)

Valencia, November 27, 2014

Based on a joint work with Loukas Grafakos

M. Mastyło (UAM) On interpolation of analytic families of multilinear operators 1 / 36

Outline

1 Introduction

2 Analytic families of multilinear operators

3 Main results

4 Applications to Lorentz spaces

5 Applications to Hardy spaces

6 An application to the bilinear Bochner-Riesz operators


Introduction

∙ Stein’s interpolation theorem [Trans. Amer. Math. Soc. (1956)] for analyticfamilies of operators between Lp spaces (p 1) has found several significantapplications in harmonic analysis. This theorem provides a generalization ofthe classical single-operator Riesz -Thorin interpolation theorem to a family{Tz} of operators that depend analytically on a complex variable z .

∙ In the framework of Banach spaces, interpolation for analytic families ofmultilinear operators can be obtained via duality in a way similar to that usedin the linear case. For instance, one may adapt the proofs in Zygmund bookand Berg and Lofstrom for a single multilinear operator to a family ofmultilinear operators. However, this duality-based approach is not applicableto quasi-Banach spaces since their topological dual spaces may be trivial.


Introduction

The open strip {z ; 0 < Re z < 1} in the complex plane is denoted by S, itsclosure by S and its boundary by ∂S.

Definition Let A(S) be the space of scalar-valued functions, analytic in S andcontinuous and bounded in S. For a given couple (A0,A1) of quasi-Banachspaces and A another quasi-Banach space satisfying A ⊂ A0 ∩ A1, we denoteby F(A) the space of all functions f : S → A that can be written as finitesums of the form

f (z) =N∑

k=1ϕk(z)ak , z ∈ S,

where ak ∈ A and ϕk ∈ A(S). For every f ∈ F(A) we set

‖f ‖F(A) = max{

supt∈ℝ‖f (it)‖A0 , sup

t∈ℝ‖f (1 + it)‖A1

}.


Introduction

Remark Clearly we have that ‖a‖θ ¬ ‖a‖A0∩A1 for every a ∈ A0 ∩ A1, andnotice that ‖ · ‖θ could be identically zero.

Definition A quasi-Banach couple is said to be admissible whenever ‖ · ‖θis a quasi-norm on A0 ∩ A1, and in this case, the quasi-normed space(A0 ∩ A1, ‖ · ‖θ) is denoted by (A0,A1)θ.

Remark If A is dense in A0 ∩ A1, then for every a ∈ A we have

‖a‖θ = inf{‖f ‖F(A); f ∈ F(A), f (θ) = a}.

Definition If there is a completion of (A0,A1)θ which is set-theoreticallycontained in A0 + A1, then it is denoted by [A0,A1]θ.


Introduction

Theorem [A. P. Calderón, Studia Math. (1964)] If (A0,A1) is a Banachcouple, then for every f ∈ F(A0 ∩ A1), and 0 < θ < 1,

log ‖f (θ)‖θ ¬∫ ∞−∞

log ‖f (it)‖A0P0(θ, t) dt +

∫ ∞−∞

log ‖f (1 + it)‖A1P1(θ, t) dt,

where P0(θ, t) and P1(θ, t) are the values of the Poisson kernels of the strip,on Re z = 0 and Re z = 1, respectively.

Remark The same estimate holds in the case of quasi-Banach spaces; theproof is similar as in the Banach case.

∙ The Poisson kernels Pj (j = 0, 1) for the strip are given by

Pj(x + iy , t) =e−π(t−y) sinπx

sin2 πx + (cos πx − (−1)je−π(t−y))2 , x + iy ∈ S.


Introduction

Using the fact that the Poisson kernels satisfy∫ℝ

P0(θ, t) dt = 1− θ,∫ℝ

P1(θ, t) dt = θ

together with Calderón’s inequality and Jensen’s inequality, and the concavity ofthe logarithmic function, we obtain the following result:

Lemma Let (A0,A1) be a couple of complex quasi-Banach spaces. For every f inF(A0 ∩ A1), 0 < p0, p1 <∞, and 0 < θ < 1 we have

‖f (θ)‖θ ¬( 1

1− θ

∫ ∞−∞‖f (it)‖p0

A0P0(θ, t) dt

) 1−θp0(1θ

∫ ∞−∞‖f (1 + it)‖p1

A1P1(θ, t) dt

) θp1.


Analytic families of multilinear operators

Definition A continuous function F : S → ℂ which is analytic in S is said tobe of admissible growth if there is 0 ¬ α < π such that

supz∈S

log |F (z)|eα|Im z| <∞.

Lemma [I. I. Hirchman, J. Analyse Math. (1953)] If a function F : S → ℂ isanalytic, continuous on S, and is of admissible growth, then

log |F (θ)| ¬∫ ∞−∞

log |F (it)|P0(θ, t) dt +

∫ ∞−∞

log |F (1 + it)|P1(θ, t) dt.



Definition Let (Ω,Σ, µ) be a measure space and let X1,...,Xm be linearspaces. The family {Tz}z∈S of multilinear operatorsT : X1 × · · · × Xm → L0(µ) is said to be analytic if for any(x1, ..., xm) ∈ X1 × · · · × Xm and for almost every ω ∈ Ω the function

z 7→ Tz (x1, ..., xm)(ω), z ∈ S

is analytic in S and continuous on S. Additionally, if for j = 0 and j = 1 thefunction

(t, ω) 7→ Tj+it(x1, ..., xm)(ω), (t, ω) ∈ ℝ× Ω (∗)

is (L × Σ)-measurable for every (x1, ..., xm) ∈ X1 × · · · × Xm, and for almostevery ω ∈ Ω the function given by formula (∗) is of admissible growth, thenthe family {Tz} is said to be an admissible analytic family. Here L is theσ-algebra of Lebesgue measurable sets in ℝ.



∙ A quasi-Banach lattice X is said to be maximal whenever 0 ¬ fn ↑ f a.e.,fn ∈ X , and supn1 ‖fn‖X <∞ implies that f ∈ X and ‖fn‖X → ‖f ‖X .

∙ A quasi-Banach lattice X is said to be p-convex (0 < p <∞) if there existsa constant C > 0 such that for any f1,...,fn ∈ X we have∥∥∥( n∑

k=1|fk |p

)1/p∥∥∥X¬ C

( n∑k=1‖fk‖p

X

)1/p.

The optimal constant C in this inequality is called the p-convexity constantof X , and is denoted, by M(p)(X ).

∙ A quasi-Banach lattice is said to have nontrivial convexity whenever it isp-convex for some 0 < p <∞.


Main results

Theorem For each 1 ¬ i ¬ m, let X i = (X0i ,X1i ) be admissible couples ofquasi-Banach spaces, and let (Y0,Y1) be a couple of maximal quasi-Banachlattices on a measure space (Ω,Σ, µ) such that each Yj is pj -convex forj = 0, 1. Assume that Xi is a dense linear subspace of X0i ∩ X1i for each1 ¬ i ¬ m, and that {Tz}z∈S is an admissible analytic family of multilinearoperators Tz : X1 × · · · × Xm → Y0 ∩ Y1. Suppose that for every(x1, ..., xm) ∈ X1 × · · · × Xm, t ∈ ℝ and j = 0, 1,

‖Tj+it(x1, ..., xm)‖Yj ¬ Kj(t)‖x1‖Xj1 · · · ‖xm‖Xjm ,

where Kj are Lebesgue measurable functions such that Kj ∈ Lpj (Pj(θ, ·) dt)

for all θ ∈ (0, 1). Then for all (x1, ..., xm) ∈ X1 × · · · × Xm, all s ∈ ℝ, and all0 < θ < 1 we have

‖Tθ+is(x1, ..., xm)‖Y 1−θ0 Y θ1

¬ (M(p0)(Y0))1−θ

(M(p1)(Y1))θKθ(s)

m∏i=1‖xi‖(X0i ,X1i )θ ,

where

log Kθ(s) =

∫ℝ

P0(θ, t) log K0(t + s) dt +

∫ℝ

P1(θ, t) log K1(t + s) dt.


Main results

Lemma Let (X0,X1) be a couple of complex quasi-Banach lattices ona measure space (Ω,Σ, µ) such that X0 is p0-convex and X1 is p1-convex.Then for every 0 < θ < 1 we have

‖x‖X 1−θ0 Xθ1

¬ (M(p0)(X0))1−θ(M(p1)(X1))θ ‖x‖(X0,X1)θ , x ∈ X0 ∩ X1.

In particular (X0,X1) is an admissible quasi-Banach couple.

Lemma Let (X0,X1) be a couple of complex quasi-Banach lattices ona measure (Ω,Σ, µ). If xj ∈ Xj are such that |xj | (j = 0, 1) are boundedabove and their non-zero values have positive lower bounds, then

|x0|1−θ|x1|θ ∈ (X0,X1)θ

and ∥∥|x0|1−θ|x1|θ∥∥

(X0,X1)θ¬ ‖x0‖1−θ

X0‖x1‖θX1

.


Main results

Corollary Let (X0,X1) be a couple of complex quasi-Banach lattices ona measure space (Ω,Σ, µ). If x ∈ X0 ∩ X1 has an order continuous norm inX 1−θ

0 X θ1 , then for every 0 < θ < 1,

‖x‖(X0,X1)θ ¬ ‖x‖X 1−θ0 Xθ1

.

Theorem Let (X0,X1) be a couple of complex quasi-Banach lattices ona measure space with nontrivial lattice convexity constants. If the spaceX 1−θ

0 X θ1 has order continuous quasi-norm, then

[X0,X1]θ = X 1−θ0 X θ

1

up equivalences of norms (isometrically, provided that lattice convexityconstants are equal to 1). In particular this holds if at least one of the spacesX0 or X1 is order continuous.


Main results

Theorem For each 1 ¬ i ¬ m, let (X0i ,X1i ) be complex quasi-Banachfunction lattices and let Yj be complex pj -convex maximal quasi-Banachfunction lattices with pj -convexity constants equal 1 for j = 0, 1. Supposethat either X0i or X1i is order continuous for each 1 ¬ i ¬ m. Let T bea multilinear operator defined on (X01 + X11)× · · · × (X0m + X1m) andtaking values in Y0 + Y1 such that

T : Xi1 × · · · × Xim → Yi

is bounded with quasi-norm Mi for i = 0, 1. Then for 0 < θ < 1,

T : (X01)1−θ(X11)θ × · · · × (X0m)1−θ(X1m)θ → Y 1−θ0 Y θ

1

is bounded with the quasi-norm

‖T‖ ¬ M1−θ0 Mθ

1 .


Main results

As an application we obtain the following interpolation theorem for operatorswas proved by Kalton (1990), which was applied to study a problem inuniqueness of structure in quasi-Banach lattices (Kalton’s proof uses a deeptheorem by Nikishin and the theory of Hardy Hp-spaces on the unit disc).

Theorem Let (Ω1,Σ1, µ1) and (Ω2,Σ2, µ2) be measures spaces. Let Xi ,i = 0, 1, be complex pi -convex quasi-Banach lattices on (Ω1,Σ1, µ1) and letYi be complex pi -convex maximal quasi-Banach lattices on (Ω2,Σ2, µ2) withpi -convexity constants equal 1. Suppose that either X0 or X1 is ordercontinuous. Let T : X0 + X1 → L0(µ2) be a continuous operator such thatT (X0) ⊂ Y0 and T (X1) ⊂ Y1. Then for 0 < θ < 1,

T : X 1−θ0 X θ

1 → Y 1−θ0 Y θ

1

and‖T‖X 1−θ

0 Xθ1→Y 1−θ0 Y θ1

¬ ‖T‖1−θX0→Y0

‖T‖θX1→Y1.


Applications to Lorentz spaces

The Lorentz space Lp,q := Lp,q(Ω) on a measure space (Ω,Σ, µ),0 < p <∞, 0 < q ¬ ∞ consists of all f ∈ L0(µ) such that

‖f ‖Lp,q :=

(∫ ∞

0

(t1/pf ∗(t)

)q dtt

)1/qif 0 < q <∞

supt>0 t1/pf ∗(t) if q =∞ .

∙ The decreasing rearrangement f ∗ of f with respect to µ is defined by

f ∗(t) = inf{s > 0; µf (s) ¬ t}, t 0,

whereµf (s) = µ({ω ∈ Ω; |f (ω)| > s}), s > 0.



Lemma Let 0 < pj , qj <∞ and let Lpj ,qj for j = 0, 1 be Lorentz spaces on aninfinite nonatomic measure space (Ω,Σ, µ). Then for 0 < θ < 1 thequasi-norm of

Xθ := (Lp0,q0 )1−θ(Lp1,q1 )θ

is equivalent to that of Lp,q, where 1/p = (1− θ)/p0 + θ/p1 and1/q = (1− θ)/q0 + θ/q1. Moreover for all f ∈ Xθ we have

2−1/p ‖f ‖Lp,q ¬ ‖f ‖Xθ ¬21/p

(log 2)s ss(p1−θ0 pθ1 )s ‖f ‖Lp,q ,

where s = 1 whenever 1 < p0, p1 <∞ and 1 ¬ q0, q1 <∞ ands > max{1/p0, 1/q0, 1/p1, 1/q1} otherwise.



Theorem Let (Ω1,Σ1, µ1) and (Ω2,Σ2, µ2) be measure spaces. For1 ¬ i ¬ m, fix 0 < q0, q1, q0i , q1i <∞, 0 < r0, r1, r0i , r1i ¬ ∞ and for0 < θ < 1, define q, r , qi , ri by setting

1qi

=1− θq0i

+θ

q1i,

1ri

=1− θ

r0i+

θ

r1i,

1q =

1− θq0

+θ

q1,

1r =

1− θr0

+θ

r1.

Assume that Xi is a dense linear subspace of Lq0i ,r0i (Ω1) ∩ Lq1i ,r1i (Ω1) andthat {Tz}z∈S is an admissible analytic family of multilinear operatorsTz : X1 × · · · × Xm → Lq0,r0 (Ω2) ∩ Lq1,r1 (Ω2). Suppose that for every(h1, ..., hm) ∈ X1 × · · · × Xm, t ∈ ℝ and j = 0, 1, we have

‖Tj+it(h1, ..., hm)‖Lqj ,rj (Ω2) ¬ Kj(t)‖h1‖Lqj1,rj1 (Ω1) · · · ‖hm‖Lqjm,rjm (Ω1)


for all θ ∈ (0, 1), where pj is chosen so that 0 < pj < qj and pj ¬ rj for eachj = 0, 1.



Then for all (f1, ..., fm) ∈ X1 × · · · × Xm, 0 < θ < 1, and s ∈ ℝ we have

‖Tθ+is(f1, ..., fm)‖(Lq0,r0 ,Lq1,r1 )θ ¬ Cθ Kθ(s)m∏

i=1‖fi‖(Lq0,r0 ,Lq1,r1 )θ ,

whereC(θ) :=

( q0q0 − p0

) 1−θp0( q1

q1 − p1

) θp1,

log Kθ(s) =

∫ℝ


∫ℝ




If in addition the measures spaces are infinite and nonatomic, then for all(f1, ..., fm) in X1 × · · · × Xm and s ∈ ℝ, and 0 < θ < 1 we have

‖Tθ+is(f1, ..., fm)‖Lq,r (Ω2) ¬ C( q0

q0 − p0

) 1−θp0( q1

q1 − p1

) θp1 Kθ(s)

m∏i=1‖fi‖Lqi ,ri (Ω1),

whereC = 2

1q +∑m

i=11qi

(u q1−θ

0 qθ1log 2

)u

with u = 1 if 1 < q0, q1 <∞ and 1 ¬ r0, r1 ¬ ∞, whileu > max{1/q0, 1/q1, 1/r0, 1/r1} otherwise.


Applications to Hardy spaces

Suppose that there is an operator M defined on a linear subspace of L0(Ω,Σ, µ)

and taking values in L0(Ω,Σ, µ) such that:

(a) For j = 0 and j = 1 the function (t, x) 7→ M(h(j + it, ·))(ω), (t, ω) ∈ ℝ× Ω

is L × Σ-measurable for any function h : ∂S × Ω→ ℂ such thatω 7→ h(j + it, ω) is Σ-measurable for almost all t ∈ ℝ.

(b) M(λh)(ω) = |λ|M(h)(ω) for all λ ∈ ℂ.(c) For every function h as in above there is an exceptional set Eh ∈ Σ with

µ(Eh) = 0 such that for j ∈ {0, 1}

M(∫ ∞−∞

h(t, ·)Pj(θ, t) dt)

(ω) ¬∫ ∞−∞M(h(t, ·))(ω)Pj(θ, t) dt

for all z ∈ ℂ, all θ ∈ (0, 1), and all ω /∈ Eh. Moreover, Eψh = Eh for everyanalytic function ψ on S which is bounded on S.



For each 1 ¬ i ¬ m, let X i = (X0i ,X1i ) be admissible couples ofquasi-Banach spaces, and let (Y0,Y1) be a couple of complex maximalquasi-Banach lattices on a measure space (Ω,Σ, µ) such that each Yj ispj -convex for j = 0, 1. Assume that Xi is a dense linear subspace of X0i ∩ X1ifor each 1 ¬ i ¬ m, and that {Tz}z∈S is an admissible analytic family ofmultilinear operators Tz : X1 × · · · × Xm → Y0 ∩ Y1. Assume that M isdefined on the range of Tz , takes values in L0(Ω,Σ, µ), and satisfiesconditions (a), (b) and (c).



Theorem Suppose that for every (x1, ..., xm) ∈ X1 × · · · × Xm, t ∈ ℝ and

‖M(Tj+it(x1, ..., xm))‖Yj ¬ Kj(t)‖x1‖Xj1 · · · ‖xm‖Xjm , j = 0, 1,


for all θ ∈ (0, 1). Then for all (x1, ..., xm) ∈ X1 × · · · × Xm, s ∈ ℝ, and0 < θ < 1,

‖M(Tθ+is(x1, ..., xm))‖Y 1−θ0 Y θ1

¬ Cθ Kθ(s)m∏

i=1‖xi‖(X0i ,X1i )θ ,

whereCθ = (M(p0)(Y0))

1−θ(M(p1)(Y1))

θ,

log Kθ(s) =

∫ℝ


∫ℝ




The preceding theorem has an important application to interpolation ofmultilinear operators that take values in Hardy spaces. A particular case ofthe above Theorem arises when:

∙ Ω = ℝn, µ is Lebesgue measure, and

M(h)(x) = supδ>0|φδ ∗ h(x)|, x ∈ ℝn

where φ is a Schwartz function on ℝn with nonvanishing integral.

∙ Y0 = Lp0 , Y1 = Lp1 , in which case Y 1−θ0 Y θ

1 = Lp, where1/p = (1− θ)/p0 + θ/p1.

Definition The classical Hardy space Hp of Fefferman and Stein is defined by

‖h‖Hp := ‖M(h)‖Lp .



Corollary If {Tz} is an admissible analytic family is such that

‖Tj+it(x1, ..., xm)‖Hpj ¬ Kj(t)‖x1‖Xj1 · · · ‖xm‖Xjm , j = 0, 1,

then

‖Tθ+s(x1, ..., xm)‖Hp ¬ Kθ(s)m∏

i=1‖xi‖(X0i ,X1i )θ

for 0 < p0, p1 <∞, s ∈ ℝ, and 0 < θ < 1. Analogous estimates hold for theHardy-Lorentz spaces Hq,r where estimates of the form

‖Tj+it(x1, ..., xm)‖Hqj ,rj ¬ Kj(t)‖x1‖Xj1 · · · ‖xm‖Xjm

for admissible analytic families {Tz} when j = 0, 1 imply

‖Tθ+is(x1, ..., xm)‖Hq,r ¬ C Kθ(s)m∏

i=1‖xi‖(X0i ,X1i )θ ,

whereC := 2

1q

(u q1−θ

0 qθ1log 2

)u( q0q0 − p0

) 1−θp0( q1

q1 − p1

) θp1,

0 < pj < qj <∞, pj ¬ rj ¬ ∞ and 1/q = (1− θ)/q0 + θ/q1,1/r = (1− θ)/r0 + θ/r1 while u = 1 if 1 < q0, q1 <∞ and 1 ¬ r0, r1 ¬ ∞and u > max{1/q0, 1/q1, 1/r0, 1/r1} otherwise.M. Mastyło (UAM) On interpolation of analytic families of multilinear operators 30 / 36

An application to the bilinear Bochner-Riesz operators

Stein’s motivation to study analytic families of operators might have been thestudy of the Bochner-Riesz operators

Bδ(f )(x) :=

∫|ξ|¬1

(1− |ξ|2

)δ f (ξ)e2πix ·ξdξ.

in which the “smoothness” variable δ affects the degree p of integrability ofBδ(f ) on Lp(ℝn). Here f is a Schwartz function on ℝn and f is its Fouriertransform defined by

f (ξ) =

∫ℝn

f (x)e−2πix ·ξdx , ξ ∈ ℝn.

Remark Using interpolation for analytic families of operators, Stein showedthat whenever δ > (n − 1)|1/p − 1/2|, then

Bδ : Lp(ℝn)→ Lp(ℝn)

is bounded for every 1 ¬ p ¬ ∞.



∙ The bilinear Bochner-Riesz operators are defined on S × S by

Sδ(f , g)(x) :=

∫∫|ξ|2+|η|2¬1

(1− |ξ|2 − |η|2

)δ f (ξ)g(η)e2πix ·(ξ+η)dξdη

for every f , g ∈ S.

∙ The bilinear Bochner-Riesz means Sz is defined by

Sz (f , g)(x) =

∫ ∫Kz (x − y1, x − y2)f (y1)g(y2)dy1 dy2,

where that the kernel of Sδ+it is given by

Kδ+it(x1, x2) =Γ(δ + 1 + it)

πδ+itJδ+it+n(2π|x |)|x |δ+it+n , x = (x1, x2).



If δ > n − 1/2, then using known asymptotics for Bessel functions we havethat this kernel satisfies an estimate of the form:

|Kδ+it(x1, x2)| ¬ C(n + δ + it)

(1 + |x |)δ+n+1/2 ,

where C(n + δ + it) is a constant that satisfies

C(n + δ + it) ¬ Cn+δeB |t|2

for some B > 0 and so we have

|Kδ+it(x1, x2)| ¬ Cn+δ eB|t|2 1(1 + |x1|)n+ε

1(1 + |x2|)n+ε

,

with ε = 12 (δ − n − 1/2). It follows that the bilinear operator Sδ+it is

bounded by a product of two linear operators, each of which has a goodintegrable kernel. It follows that

K δ+it : L1 × L1 → L1/2

with constant K1(t) ¬ C ′n+δeB|t|2 whenever δ > n − 1/2.M. Mastyło (UAM) On interpolation of analytic families of multilinear operators 34 / 36


Theorem Let 1 < p < 2. For any λ > (2n − 1)(1/p − 1/2)

Sλ : Lp(ℝn)× Lp(ℝn)→ Lp/2(ℝn) is bounded .

Proof. We apply the main Theorem with X01 = X11 = L2, X02 = X12 = L1,Y0 = L1, Y1 = L1/2, Xi is the space of Schwartz functions on ℝn, which isdense in L1 and L2. We fix δ > 0 and we consider the bilinear analytic family{Tz}z∈S , where Tz := S(n− 1

2 )z+δ for all z ∈ S. We claim that this family isadmissible. Indeed, for f , g Schwartz functions we have

Tz (f , g)(x) =

∫∫|ξ|2+|η|2¬1

(1− |ξ|2 − |η|2

)(n− 12 )z+δ f (ξ)g(η)e2πix ·(ξ+η)dξdη, x ∈ ℝn,

and the map z 7→ Tz (f , g) is analytic in S, continuous and bounded on S,and jointly measurable in (t, x) when z = it or z = 1 + it. Moreover, for allx ∈ ℝn we have

supz∈S

log |Tz (f , g)(x)|eα|Im z| <∞, f , g ∈ S

with α = 0 < π; in fact |Tz (f , g)(x)| ¬ ‖f ‖L1‖g ‖L1 .M. Mastyło (UAM) On interpolation of analytic families of multilinear operators 35 / 36


Based on the preceding discussion, we have that when Re z = 0, Tz mapsL2 × L2 to L1 with constant K0(t) ¬ Cn,δec |t|2 for some Cn,δ, c > 0. We alsohave that when Re z = 1, Tz maps L1 × L1 to L1/2 with constantK1(t) ¬ C ′n,δeB |t|2 for some C ′n,δ, B > 0. We notice that for these functionsKi (t) we have that the constant K (θ, 1, 1/2) <∞; in this case θ = 2( 1

p −12 ).

An application of the main Theorem yields that

Sλ : Lp(ℝn)× Lp(ℝn)→ Lp/2(ℝn)

provided λ = 2(n − 12 ) ( 1

p −12 ) + δ > (2n − 1)( 1

p −12 ).


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