Operator-valued Lp-Lq-Fourier multipliers · Operator-valued Lp-Lq-Fourier multipliers Mark Veraar...
Transcript of Operator-valued Lp-Lq-Fourier multipliers · Operator-valued Lp-Lq-Fourier multipliers Mark Veraar...
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
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 2 / 16
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)
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 3 / 16
Introduction: What is a Fourier multiplier?
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....
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 4 / 16
Introduction: What is a Fourier multiplier?
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
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 5 / 16
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
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 5 / 16
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
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 6 / 16
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
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 6 / 16
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.
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 7 / 16
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
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 8 / 16
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
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 9 / 16
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.
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 10 / 16
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.
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 11 / 16
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.
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 12 / 16
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
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 12 / 16
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.
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 13 / 16
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.
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 14 / 16
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 .
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 15 / 16
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/
Mark Veraar (TU Delft) Lp -Lq -Fourier multipliers Bedlewo, April 2017 16 / 16