Weighted Littlewood-Paley inequalities

Nicolaus Copernicus University, Toruń, Poland 3-7 June 2013, Herrnhut, Germany

for every interval I in R let S I denote the partial sum operator, i.e., (S I f ) = χ I f (f L2 (R)), for an arbitrary family I of disjoint intervals in R let S I denote the Littlewood-Paley square function, i.e., ( ) 1/2 S I f := S I f 2 I I (f L 2 (R)). Theorem (Littlewood-Paley; 1926) Let I be the dyadic decomposition of R, i.e., I := { ±(2 j, 2 j+1 ] : j Z }. Then, S I is bounded on L p (R) for every p (1, ).

A p (R) the class of weights on R which satisfy Muckenhoupt s A p -condition (p [1, )): w A p(r) (1 < p < ) if ( 1 sup a<b b a w A 1(R), if b a ) ( 1 w b a b a ) p 1 w 1 p =: [w] Ap < Mw C w for some constant C > 0, [w] A1 := inf C Theorem (Kurtz; 1980) If I is the dyadic decomposition of R, then the square function S I is bounded on L p (R, wdx) for every 1 < p < and every w A p (R).

Theorem (Rubio de Francia s inequalities; 1985) For an arbitrary family I of disjoint intervals in R the square function S I is bounded on L p (R, wdx) for every 2 < p < and w A p/2 (R).

Theorem (Rubio de Francia s inequalities; 1985) For an arbitrary family I of disjoint intervals in R the square function S I is bounded on L p (R, wdx) for every 2 < p < and w A p/2 (R). Rubio de Francia s conjecture For an arbitrary family I of disjoint intervals in R the square function S I is bounded on L 2 (R, wdx) for every Muckenhoupt weight w A 1 (R). K.,, in preparation Theorem There exists a constant C > 0 such that for any family I of disjoint intervals in R S I f C [w]3 L2(R,wdx) A 1 f L2(R,wdx) (f L 2 (R, wdx)) for every Muckenhoupt weight w A 1 (R).

Theorem (Rubio de Francia s extrapolation theorem; 1982) Let 1 λ r <, and let S be a family of sublinear operators which is uniformly bounded in L r (w) for each w A r/λ (R n ), i.e., Sf r wdx C r,w f r wdx (S S, w A r/λ (R n )). If λ < p, q < and w A p/λ (R n ), then S is uniformly bounded in L p (w) and even more: ( S j f j q ) p/q wdx C p,q,w ( f j q ) p/q wdx j j for all f j L p (w), S j S.

Theorem (a variant of Rubio de Francia s extrapolation theorem; 1982) Let 1 r <, and let S be a sublinear operator which is bounded in L r (w) for each w A r (R n ). Then S is bounded on L p (w) for every 1 < p < and w A p (R n ). The philosophy underlying the extrapolation theory developed by Rubio de Francia can be summarized as follows: A. Córdoba (1987): There are no L p spaces, only weighted L 2 J.-L. Rubio de Francia (1987): The boundedness properties of a linear operator depend only on the weighted L 2 inequalities that it satisfies

Theorem (Rubio de Francia s inequalities; 1985) For an arbitrary family I of disjoint intervals in R the square function S I is bounded on L p (R, wdx) for every 2 < p < and w A p/2 (R). Approach The proof is based on the Calderón-Zygmund theory for operator-valued kernels, which was systematically studied in : J. Rubio de Francia, F. J. Ruiz and J. L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Adv. in Math. 62 (1986), 7 48. Crucial steps: the Hilbert space-valued kernel related to a smooth version of S I, G, satisfies weak-(d 2 ) condition this leads to the following pointwise estimate for G: M (Gf )(x) C M( f 2 )(x) 1/2 for a.e. x R, f L c (R),

an alternative proof of Rubio de Francia s inequalities another proof was given by J. Bourgain (1985); an alternative proof of pointwise estimate was given by P. Sjölin (1986). the extension of unweighted variant of Rubio de Francia s inequalities to higher dimensions is due to J.-L. Journé (1985). simpler arguments were later given by F. Soria (1987) in two dimensions, and in higher dimensions by S. Sato (1990) and X. Zhu (1991). recently, M. Lacey (2007) proposed another approach based on time frequency arguments.

J. Rubio de Francia, F. J. Ruiz and J. L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Adv. in Math. 62 (1986), 7 48. according to Part IV(E) of : the A 1 -weighted L 2 -estimates stated in Conjecture can be reached by interpolation provided that I is a family such that S I is bounded on (unweighted) L p (R) for some p < 2. a variant of the Littlewood-Paley decomposition For which partitions I of R a local variant of the Littlewood-Paley decomposition theorem holds, i.e., there exists r 2 such that S I bounded on L p (R) for all 1 p 1 2 < 1 r. is

Lack of a local variant of the Littlewood-Paley property L. Carleson (1967); for I := {[n, n + 1) : n Z}, proved that S I is bounded on L p (R) if p 2, and unbounded on L p (R) if p < 2. R. Edwards, G. Gaudry (1977): a = (a j ) j=0 (0, ) is an increasing sequence, the decomposition I of R determined by a, i.e., I a := { ( a 0, a 0 ) } { ±[a j 1, a j ) } j 1 if a j+1 a j j m as j, then S Ia is unbounded on L p(r) for every p < 2; a = (a j ) j=0 increasing sequence such that a j+1 a j 2 φ(j)j, where φ(j) 0 + arbitrary slowly as j, the square function S Ia is not bounded on L p (R) for every p < 2.

Main result & Sketch of the proof Theorem There exists a constant C > 0 such that for any family I of disjoint intervals in R and for every Muckenhoupt weight w A 1 (R). S I f C [w]3 L2(R,wdx) A 1 f L2(R,wdx) (f L 2 (R, wdx)) Approach The proof is based on the first ideas centering around Carleson s result by A. Córdoba (1979 and 1981); (slight generalizations was given) by J.-L. Rubio de Francia (1983), who proved: for families of so-called almost congruent intervals; (by combining this result with Kurtz one) for families determined by slowly increasing convex sequences applying his arguments one can produce a huge variety of configurations of intervals for which A 1 -weighted L 2 -estimates would turn out to be true, but it seems that the general case has been out of reach by such techniques

Sketch of the proof Lemma S I 2,w := sup{ S I f 2,w : f L 2 (R, wdx), f 2,w = 1} [0, ] by a decomposition of an interval I in R we mean any family J of intervals such that J J J = I There exists a constant C > 0 such that for any family I of disjoint intervals in R and arbitrary decompositions J I of intervals I (I I): ( ) S I 2,w C sup S J I 2,w + [w] A1 S J 2,w I I for every Muckenhoupt weight w A 1 (R), where J := I I J I. Proof of Lemma. (1) There exists a constant C > 0 such that for every interval I in R and every Muckenhoupt weight w A 1 (R) S I 2,w C [w] A1

Sketch of the proof Lemma There exists a constant C > 0 such that for any family I of disjoint intervals in R and arbitrary decompositions J I of intervals I (I I): ( ) S I 2,w C sup S J I 2,w + [w] A1 S J 2,w I I for every Muckenhoupt weight w A 1 (R), where J := I I J I. Proof of Lemma. (1) There exists a constant C > 0 such that for every interval I in R and every Muckenhoupt weight w A 1 (R) S I 2,w C[w] A1 (2) Fix w A 1 (R) and I I. Let JI denote the complement of J I to the decomposition of R by adding connected components of R \ I (3) Thus S J I 2,w C( S J I 2,w + [w] A1 ), where C is independent of I and J I.

Sketch of the proof (1) There exists a constant C > 0 such that for every interval I in R and every Muckenhoupt weight w A 1 (R) S I 2,w C[w] A1 (2) Fix w A 1 (R) and I I. Let JI denote the complement of J I to the decomposition of R by adding connected components of R \ I (3) Thus S J I 2,w C( S J I 2,w + [w] A1 ), where C is independent of I and J I. (4) J J S I J f converges to f in L 2 (w). (5) By the Cauchy-Schwarz inequality and the converse of Hölder inequality we get S I 2,w S J I 2,w S J I 2,w.

Sketch of the proof we say that a family I consists of almost congruent intervals (with constant 2) if sup I 2 inf I, I I I I where I denotes the length of I.

Sketch of the proof we say that a family I consists of almost congruent intervals (with constant 2) if sup I 2 inf I, I I I I where I denotes the length of I. Lemma (Rubio de Francia; 1983) There exists a constant C > 0 such that for any family I of almost congruent intervals in R and for every Muckenhoupt weight w A 1 (R) S I 2,w C[w] 3/2 A 1.

Sketch of the proof Proof of Lemma. (1) Note that we can assume that 1 I 2 for all I I Indeed, if a := sup I I I 2 b := 2 inf I I I, then the family I := κi = {κi : I I}, where κ = 2 and κi := {κx : x I }, a satisfies this specific assumption. Then, for every I I, w A 1(R), and f L 2(w), S I f (x) = (S κi f (κ )) ( /κ), which yields S I 2,w C[w( /κ)] 3/2 A 1. It is easy to see that [w] A1 = [w( /κ)] A1 for every w A 1(R). (2) Let φ be a Schwartz function such that ˆφ(ξ) = 1 on ξ [ 2, 2]. Fix n I I Z for every I I and set φ I := e 2πin I ( ) φ. Consider the smooth version of S I, G I, given by ( ) 1/2 G I f := φ I f 2 I I (f L 2 (R)).

Sketch of the proof Proof of Lemma. (3) There exists a constant C φ depending only on φ, such that Gf (x) 2 w(x)dx C φ f (x) 2 Mw(x)dx C φ [w] A1 f (x) 2 w(x)dx R R R for every f L 2 (w).

Sketch of the proof Proof of Lemma. (3) There exists a constant C φ depending only on φ, such that Gf (x) 2 w(x)dx C φ f (x) 2 Mw(x)dx C φ [w] A1 f (x) 2 w(x)dx R for every f L 2 (w). R (4) There exists a constant C > 0 such that the following vector inequality holds: ( S I f I 2 ) 1/2 C[w] A1 ( f I 2 ) 1/2 I I 2,w I I 2,w for every weight w A 1 (R) and for every (f I ) I I (for which the left hand side is finite). R

Sketch of the proof Proof of Theorem. Fix I and w A 1 (R). Let I n := {I I : I > 1 n }, n N. We show that the square functions S In are uniformly bounded on L 2 (w).

Sketch of the proof Proof of Theorem. Fix I and w A 1 (R). Let I n := {I I : I > 1 n }, n N. We show that the square functions S In are uniformly bounded on L 2 (w). Set I n =: {I n,k } k=1,...,kn, where k n N { }, Let J n,k, k {1,..., k n } be a decomposition of I n,k on j n,k := [n I n,k ] subintervals such that j n,k 1 of them have lengths equal 1 n. Note that the families J n := k J n,k and J n,k, k = 1,..., k n, consist of almost congruent intervals.