SINGULAR INTEGRALS WITH ANGULAR INTEGRABILITY

Transcription

1 SINGULAR INTEGRALS WITH ANGULAR INTEGRABILITY FEDERICO CACCIAFESTA AND RENATO LUCÀ Abstract. In this note we rove a class of shar inequalities for singular integral oerators in weighted Lebesgue saces with angular integrability. arxiv: v [math.ap] 5 Feb 6. Introduction We consider singular integral oerators Tf(x : P.V. f(x yk(y dy, (. R n where the kernel K satisfies the following conditions y n K C, y n+ K C, K C. Here C > is a constant and denotes the Fourier transform. The main examle we have in mind is the directional Riesz transform, which corresonds to the choice K(y : y (n+ y θ, θ S n. The study of the boundedness of these oerators in weighted Lebesgue saces L (w(xdx, for < < and < w L loc (Rn, is a classical roblem in harmonic analysis: in articular, Stein [3] roved it for the (shar range of homogeneous weights w(x x α, n/ < α < n n/. The result was later extended by Coifman and Fefferman [] to any A weight. While the weighted L -theory has been extensively studied, less is known in the case of Lebesgue norms with different integrability in the radial and angular directions, namely f L x L θ : ( + f(ρ L (S n ρn dρ. (. These mixed radial-angular saces have been successfully used in recent years to imrove several results in the framework of artial differential equations; see e.g. [, 8,,,, 4, 5]. Notice that when the norms reduce to the usual L norms. Notice also that, neglecting the constants, they are increasing in, and that they behave as the L norms under homogeneous rescaling, namely f( f(λ, λ >. In a recent aer A. Córdoba [3] roved, among the other things, the L x L θ boundedness for oerators of the form (.. Here we give an extension of this result to the weighted setting. Theorem.. Let n, < <, < < and n/ < α < n n/. Then x α Tφ L C x α φ L, (.3 L where C is a constant deending only on α,,,n. Mathematics Subject Classification. 4B37, 4B. Key words and hrases. Singular integrals. Angular integrability.

2 SINGULAR INTEGRALS WITH ANGULAR INTEGRABILITY Let us oint out that the case α, < < in inequality (.3 may be deduced by the alication of Córdoba s argument; see [6, Theorem.6]. Therefore the novelty of Theorem. is in that it covers all the ossible homogeneous weights of the kind x α. Remark.. Condition n/ < α turns to be necessary by testing the inequality on functions φ such that φ and Tφ > in a neighborhood of the origin. On the other hand, condition α < n n/ turns to be necessary for the same reason by considering the dual inequality. Remark.. One of the estimates (.3 has been used in [6, Theorem.5] to deduce informations about the regularity of weak solutions of the 3d Navier Stokes roblem with initial velocities satisfying good angular integrability roerties. We write A B if A CB with a constant C deending only on α,,,n. We write A B if both A B and B A. is. Proof We know by [6, Theorem.6] that inequality (.3 is true in the case α, that Tg L g L. (. L Following Stein [3], we now show that the weighted case (.3 can be then deduced by the unweighted one. The next lemma reresents the core of the roof. Lemma.. Let n, < <,, n/ < α < n n/ and then F(x,y : ( x / y α x y n, (. F(x,yφ(ydy R n L L C φ L. (.3 L Assume indeed this has been roved and first aly inequality (. with the choice g : α f to have T( x α f L x α f L. (.4 L Then notice that T( x α f x α Tf K(x y( y α x α f(y dy R n y α x α R x y n f(y dy n ( x / y α R x y n n so that by using Lemma. with φ α f we obtain y α f(y dy, (.5 T( x α f x α Tf L x α f L. (.6 L Then, the desired estimate (.3 follows by (.4,.5 and triangle inequality. Thus it only remains to rove Lemma.. The idea is to use a change of variables which resembles the standard olar coordinates. In this variant the integration over the shere is relaced by integration over the secial orthogonal grou and the radial integration is relaced by integration over the multilicative grou of the ositive real numbers. This method works efficiently when homogeneous ower weights are involved; see e.g. [5, 7].

3 SINGULAR INTEGRALS WITH ANGULAR INTEGRABILITY 3 Proof of Lemma.. By using the isomorhism S n /SO(n we can rewrite integrals on S n as follows g(yds(y g(aeda, n, S n where da is the left Haar measure on, and e S n is a fixed unit vector. Thus, via olar coordinates, a generic integral can be rewritten as F(x,yφ(xdy F(x,ρωφ(ρωdS ω ρ n dρ R n S n F(x,ρBeφ(ρBedBρ n dρ. Hence the L θ norm can be written as F( x θ,yφ(ydy F( x Ae,yφ(ydy R n L θ (Sn R n F( x Ae, ρbeφ(ρbedb L A ( ρ n dρ L A ( where e is any fixed unit vector. We choose F as in (. and we change variables B AB in the inner integral. By the invarianceofthe measure this is equivalent to ( x /ρ β AB ( x Be ρe nφ(ρab edb ρ n dρ L A ( ( x /ρ β x Be ρe nφ(ρab edb ρ n dρ. L A ( Notice that the integral ( x /ρ β x Be ρe n φ(ρab e db G φ(a is a convolution on of the functions G(A ( x /ρβ x Ae ρe n, H(A φ(ρae. We can thus aly Young s inequality on (see for instance [9, Theorem..] to obtain, for any, the estimate F( x θ,yφ(ydy ( x /ρ β R n L θ (Sn ( x e ρθ n φ(ρθ L L θ (S n θ (Sn ρn dρ ( x /ρ β dρ ( x φ(ρθ L ρ θ n L θ (S n θ (Sn ρ (.7 where we switched back to the coordinates of S n. Then we notice F(x,yφ(ydy R n L L F( x θ,yφ(ydy ds θ x n R n L θ (Sn L (R + (,d x / x (.8

4 SINGULAR INTEGRALS WITH ANGULAR INTEGRABILITY 4 where R + ( is the moltilicative grou of ositive real numbers equied with its Haar measure dρ/ρ. Using (.7 allows to estimate (.8 with ( n x ( x /ρ β ρ ( x ρ e θ n ρ n dρ φ(ρθ L L θ (S n θ (Sn. ρ L (R + (,d x / x Notice that the inner term is a convolution on R + ( of the functions g(ρ ρ n ρ β (ρe θ n, h(ρ ρ n φ(ρθ L θ (Sn. L θ (S n Thus we can aly again Young s inequality to estimate further (.8 with g(ρ L θ (S n h(ρ L L (R + (,dρ/ρ, (.9 L (R + (,dρ/ρ θ (Sn for all. Once we have noticed that h(ρ L L (R + (,dρ/ρ φ L L, θ (Sn the concluding ste of the roof of the Lemma is reresented by showing that the first term of (.9 is bounded. We slit the integral + ρ n S n ρ β ρe θ n ds θ dρ ρ ( + ( + + and we bound searately the three terms. If < ρ < /, then ρe θ /. Thus, since ρ β < +ρ β I (ρ n +ρ n +β dρ < ( : I +II +III rovided that < and β > n/. If / ρ, we notice that ρ β ρ and we use that (see for instance Lemma. in [5] ρe θ n ds θ S ρ n to bound II ρ n ρ β dρ ρ ρ n dρ <. (. Finally, if < ρ < +, then ρe θ ρ θ ρ /. Thus, since ρ β < +ρ β III + (ρ n n +ρ n +β n dρ < rovided that > and β < n n/, that concludes the roof. Acknowledgements. The first author is suorted by the FIRB Disersive dynamics, Fourier analysis and variational methods. Part of the work was done during the first author s visit at MSRI, Berkeley (CA, within the rogram New Challenges in PDE: Deterministic Dynamics and Randomness in High and Infinite Dimensional Systems, which he acknowledges for the wonderful working conditions. The second author is suorted by the ERC grant and MINECO grant SEV--87 (Sain.

5 SINGULAR INTEGRALS WITH ANGULAR INTEGRABILITY 5 References [] F. Cacciafesta and P. D Ancona. Endoint estimates and global existence for the nonlinear Dirac equation with otential. J. Diff. Eq., 54(5:33 6, 3. [] R. R. Coifman and C. Fefferman, C. Weighted norm inequalities for maximal functions and singular integrals. Studia Math., 5:4 5, 974. [3] A. Córdoba. Singular integrals and maximal functions: the disk multilier revisited. Adv. Math., 9: 8 35, 6. [4] A. Córdoba and C. Fefferman. A weighted norm inequality for singular integrals. Studia Math., 57(:97, 976. [5] P. D Ancona and R. Lucà. Stein-Weiss and Caffarelli-Kohn-Nirenberg inequalities with higher angular integrability. J. Math. Anal. A., 388(:6 79,. [6] P. D Ancona and R. Lucà. On the regularity set and angular integrability for the Navier Stokes equation. ArXiv: [7] P. L. De Náoli, I. Drelichman and R. G. Durán. On weighted inequalities for fractional integrals of radial functions. Illinois J. Math., 55: ,. [8] D. Fang and C. Wang. Weighted Strichartz estimates with angular regularity and their alications. Forum Math., 3:8 5,. [9] Loukas Grafakos. Classical Fourier analysis, volume 49 of Graduate Texts in Mathematics. Sringer, New York, second edition, 8. [] R. Lucà. Regularity criteria with angular integrability for the Navier Stokes equation. Nonlinear Anal., 5:4 4, 4. [] S. Machihara, M. Nakamura, K. Nakanishi, and T. Ozawa. Endoint Strichartz estimates and global solutions for the nonlinear Dirac equation. J. Funct. Anal., 9(:, 5. [] T. Ozawa and K. M. Rogers Shar Morawetz estimates. J. Anal. Math., :63 75, 3. [3] E. M. Stein. Note on singular integrals. Proc. Am. Math. Soc., 8:5 54, 957. [4] J. Sterbenz. Angular regularity and Strichartz estimates for the wave equation. Int. Math. Res. Not., 4:87 3, 5. With an aendix by Igor Rodnianski. [5] T.Tao.Shericallyaveraged endointstrichartz estimates forthe two-dimensionalschr dinger equation. Comm. Partial Differential Equations 5(7-8:47 485,. Federico Cacciafesta: SAPIENZA Università di Roma, Diartimento di Matematica, Piazzale A. Moro, I-85 Roma, Italy address: cacciafe@mat.uniroma.it Renato Lucà: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Madrid, 849, Sain. address: renato.luca@icmat.es