Operator-valued Lp-Lq-Fourier multipliers · Operator-valued Lp-Lq-Fourier multipliers Mark Veraar Delft University of Technology Bedlewo, April 2017 Talk is based on joint work withJan

Operator-valued Lp-Lq-Fourier multipliers

Mark Veraar

Delft University of Technology

Bedlewo, April 2017

Talk is based on joint work with Jan Rozendaal:

Fourier multiplier theorems involving type and cotype, J. FourierAnal. Appl. 2017Fourier multiplier theorems on Besov spaces under type andcotype conditions, Banach J. of Math. Anal. 2017

Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 1 / 16

Overview

1 Introduction

2 Abstract setting

3 Multipliers and geometry of the Banach spaceMultipliers for spaces with Fourier typeMultipliers for spaces with type and cotypeConverse resultsMultipliers for positive operators on Banach lattices

4 ApplicationsX “ Y “ Lp

Schatten, functional calculus

5 Further results and open problems


Introduction: What is a Fourier multiplier?

Given m : Rd Ñ C locally integrable and of polynomial growthS - Schwartz functionsS 1 - tempered distributions

STm

ÝÝÝÝÑ S 1

F§

§

đ

İ

§

§F´1

Sm

ÝÝÝÝÑ S 1

(1)

We say m PMp,q if Tm P LpLp,Lqq, mMp,q “ TmLpLp,Lqq.M2,2 “ L8

Mp,p ãÑ L8 (converse false if p ‰ 2)



Sufficient conditions for m PMp,p in scalar case:Riesz 1924, mpξq “ signpξqMarcinkiewicz 1939, Mihlin 1956, smoothness conditions......too many papers to mention.........

Sufficient conditions for m PMp,p in vector-valued case:Burkholder 1983: mpξq “ signpξqMcConnell 1984, Bourgain 1984: Mihlin type reusltsWeis 2001, Mihlin for operator-valued multipliers...too many papers to mention....



Scalar case p,q P r1,8s studied by Hörmander 1960:If p ą q, then Mp,q “ t0u.If 1 ă p ď 2 ď q ă 8, 1

r “1p ´

1q , then Lr ãÑ Lr ,8 ãÑMp,q

..... many more results by Hörmander .......Mihlin type conditions by Lizorkin 1967

Operator-value case p ‰ q: almost nothing knowGoal: Extend Hörmander’s results to the operator-valued settingMotivation: stability of semigroupsHere the multiplier is: mpξq “ piξ ` Aq´1. Mihlin condition fails!For details: see Jan Rozendaal’s talk


1 Introduction

2 Abstract setting






Abstract setting

Given m : Rdzt0u Ñ LpX ,Y q strongly measurable + growth conditions

S pRd ; X q TmÝÝÝÝÑ S 1pRd ; Y q

F§

§

đ

İ

§

§F´1

S pRd ; X q mÝÝÝÝÑ S 1pRd ; Y q

(2)

We say m PMp,qpRd ; X ,Y q if Tm P LpLppRd ; X q,LqpRd ; Y qq,

mMp,qpRd ;X ,Y q “ TmLpLppRd ;Xq,LqpRd ;Y qq.

Aim: Find sufficient conditions for m PMp,qpRd ; X ,Y q

Operator-valued setting much more interesting than scalar case.Results typically depend on the geometry of the underlying spaces:

1 X Fourier type p and Y Fourier cotype q2 X type p and Y cotype q3 X is p-convex and Y is q-concave


1 Introduction

2 Abstract setting






Multipliers for spaces with Fourier type

X has Fourier type p if F : LppRd ; X q Ñ Lp1pRd ; X q (Peetre 1969).Connection Hausdorff–Young inequalities. Only p P r1,2s.

Original motivation: comparison of real and complex interpolation:

pX0,X1qθ,p ãÑ rX0,X1sθ ãÑ pX0,X1qθ,p1

Every X has Fourier type 1X Fourier type 2 ðñ X is a Hilbert spaceFourier type p implies Fourier type u for all u P r1,psLspΩq with s P r1,8q has Fourier type mints, s1u

Convention: X has Fourier cotype q P r2,8s if X has Fourier type q1.


Multipliers for spaces with Fourier type

PropositionAssume X has Fourier type p P r1,2s and Y has Fourier cotypeq P r2,8s and set 1

r “1p ´

1q . Then Lr pRd ;LpX ,Y qq ĎMp,qpRd ,X ,Y q.

Proof:Tmpf qLqpRd ;Y q ÀY ,q mpf Lq1 pRd ;Y q

Hölderď mp¨qLr pRd ;LpX ,Y qq

pf Lp1 pRd ;Xq

ÀX ,p mp¨qLr pRd ;LpX ,Y qqf LppRd ;Xq.

By interpolation techniques we obtain a result of Hörmander type:

TheoremAssume X has Fourier type ą p and Y has Fourier cotype ă q and set1r “

1p ´

1q . Then Lr ,8pRd ;LpX ,Y qq ĎMp,qpRd ,X ,Y q.

Open: limiting case of Theorem with Fourier type p and cotype q


Multipliers for spaces with type and cotype

Rademacher (co)type less restrictive than Fourier (co)type:LspΩq with s P r1,8q has type mints,2u and cotype maxts,2u.

TheoremAssume X has type ą p P r1,2q and Y has cotype ă q P p2,8s and set1r “

1p ´

1q . If m : Rdzt0u Ñ LpX ,Y q is strongly measurable and

t|ξ|dr mpξq : ξ P Rdzt0uu Ď LpX ,Y q

is R-bounded, then m PMp,qpRd ,X ,Y q.

“Better geometry” ùñ less decay required of m near infinity.

QuestionLimiting case of Theorem? Yes if X is p-convex and Y is q-concave.

Proofs based on results of Kalton–van Neerven–V.–Weis 2008


Converse results for given p P r1,2s, q P r2,8q

PropositionAssume that Lr pRd ;LpX ,Y qq ĎMp,qpRd ,X ,Y q with 1

r “1p ´

1q . Then

1 If Y “ C, q “ 2, then X has Fourier type p.2 If X “ C, p “ 2, then Y has Fourier cotype q.3 If Y “ X˚, q “ p1, then X has Fourier type p.

PropositionLet p P p1,2s and q P r2,8q and 1

r “1p ´

1q . Assume that for each m:

t|ξ|dr mpξq : ξ P Rdu R-bounded implies m PMp,qpRd ,X ,Y q. Then:

1 If X has cotype 2, Y “ C, q “ 2, then X has type p.2 If Y has type 2, X “ C, p “ 2, then Y has cotype q.3 If Y “ X˚ has type 2, q “ p1, then X has type p.


Converse results on R-boundedness

Write RpT q for the R-bound of T Ď LpX ,Y q.

Theorem (Clement–Prüss 2001)

Rptmpξq : ξ P Rd Lebesgue pointuq ď mMp,ppRd ,X ,Y q.

Proposition (Case 1 ă p ă q ă 8)

Let 1r “

1p ´

1q . Assume m|Qk “ mk P LpX ,Y q on disjoint normalized

cubes Qk . If m PMp,qpRd ,X ,Y q, then

Rptmk : k P Zuq ď Cp,q,dmMp,qpRd ,X ,Y q.

No general converse statements possible even in scalar case.However, examples on X “ Y “ `u can be given for whichR-boundedness of t|ξ|

dr mpξq : ξ P Rdu is necessary.


Multipliers for positive operators on Banach lattices

TheoremLet p,q P r1,8q with p ď q, and let α “ d

p ´dq . Assume X is p-convex

and Y is q-concave. Suppose that qmptq is a positive operator for eacht P Rd , and t ÞÑ qmptqx P L1pRd ; Y q for all x P X. Then

TmLp 9Hαp pRd ;Xq,LqpRd ;Y qq ď Cmp0qLpX ,Y q (3)

Proof based on Lemma by Montgomery-Smith 1996Theorem is sharp in X “ Y “ Lp X Lq

Theorem implies a very nice stability result.


1 Introduction

2 Abstract setting






Applications: X “ Y “ Lp

Fourier type theory: If 1r ą |

1p ´

1p1 |, then

m P Lr ,8pRd ;LpX ,Y qq ùñ m PMp,p1pRd ; X ,X q.

Type, cotype theory: If 1r “

ˇ

ˇ

ˇ

1p ´

12 |, then

Rpt|ξ|dr mpξq : ξ P Rdzt0uuq ùñ m PMp,qpRd ; X ,X q.


Applications: Schatten, functional calculus

C p Schatten p-class operators on `2pZq.

TheoremLet p P p1,8qzt2u and 1

r ă |1p ´

12 |. Let σ : ZÑ C be such that

Cσ :“ supjPZp1` |j |1r q|σj | ă 8, let φ : ZÑ Z and σj,k :“ mφpjq´φpkq.

Then the Schur multiplier pMaqj,k :“ σj,kaj,k satisfies MLpC pq Àp,r Cσ.

Open problem: case 1r “ |

1p ´

12 |

Theorem (Rozendaal 2015)Assume X as type p and cotype q. Let ´iA generate a C0-group. Thenf pAq : DAp

1p ´

1q ,1q Ñ X is bounded for each bounded holomorphic

function on a sufficiently large strip.


Further results and open problems

Extrapolation from m PMp,qpRd ,X ,Y q to m PMu,v pRd ,X ,Y qwith 1

u ´1v “

1p ´

1q by Mihlin type conditions

Multipliers on torus (by transference different from case p “ q)Multiplier results in the Besov scale: optimal exponentsApplications to stability theory (see next talk)

Open problems: Converses and limiting cases already mentioned.

ProblemIs there an X-valued analogue of Pitt’s inequality for 1 ă p ď q ă 8?

ξ ÞÑ |ξ|´αpf pξqLqpRd q ď Cs ÞÑ |s|βf psqLppRd q. (4)

Here α P r0, dq q, β P r0,

dp1 q and d

p `dq ` β ´ α “ d.

α “ β “ 0, connected to Fourier type of X .


Book project

Analysis in Banach spaces Volume I:Martingales and Littlewood-Paley theoryTuomas Hytönen, Jan van Neerven,Mark Veraar, Lutz Weis, 2016

Volume II: Probabilistic Techniques and Operator TheoryPreprint available on http://fa.its.tudelft.nl/~veraar/


http://fa.its.tudelft.nl/~veraar/

Documents

Operator-valued Lp-Lq-Fourier multipliers · Operator-valued Lp-Lq-Fourier multipliers Mark Veraar Delft University of Technology Bedlewo, April 2017 Talk is based on joint work withJan