We establish a uniform domination of the family of trilinear multiplier forms with singularity o... more We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one-dimensional subspace by positive sparse forms involving L^p-averages. This class includes the adjoint forms to the bilinear Hilbert transforms. Our result strengthens the L^p-boundedness proved by Muscalu, Tao and Thiele and entails as a corollary a rich multilinear weighted theory. In particular, we obtain weighted boundedness of the bilinear Hilbert transform when the weights belong to the classes A_p intersection RH 2. Our proof relies on a stopping time construction based on newly developed localized outer-L p embedding theorems for the wave packet transform.
A semilinear strongly damped wave equation with memory is considered in the past history framewor... more A semilinear strongly damped wave equation with memory is considered in the past history framework. The existence of global attractors of optimal regularity is established, both for critical and supercritical nonlinearities, under a necessary and sufficient condition on the memory kernel.
We develop a Banach-valued version of the outer Lp space theory of Do and Thiele [7], relying on ... more We develop a Banach-valued version of the outer Lp space theory of Do and Thiele [7], relying on suitable randomized analogues of the conical square func- tion and nontangential maximal function. Within this framework, we prove an ab- stract Coifman-Meyer type, operator-valued multilinear multiplier theorem for suit- able tuples of UMD spaces. A concrete case of our theorem is a multilinear gener- alization of Weis’ operator-valued Ho ̈rmander-Mihlin linear multiplier theorem [44]. Furthermore, we derive from our abstract result a wide range of mixed Lp-norm es- timates for multi-parameter multilinear multiplier operators, as well as for the more singular tensor products of a one-parameter Coifman-Meyer multiplier with a bilin- ear Hilbert transform. These respectively extend the results of Muscalu et. al. [35] and of Silva [42]. In the same setting, we derive several operator-valued T(1)-type theorems both in one parameter, and of multi-parameter, mixed-norm type. A distin- guishing feature of our T(1) theorems is that the usual explicit assumptions on the distributional kernel of T are replaced with testing-type conditions.
We study the behavior of the bilinear Hilbert transform BHT at the boundary of the known boundedn... more We study the behavior of the bilinear Hilbert transform BHT at the boundary of the known boundedness region . A sample of our results is the estimate
We prove L p (w) bounds for the Carleson operator C, its lacunary version C lac , and its analogu... more We prove L p (w) bounds for the Carleson operator C, its lacunary version C lac , and its analogue for the Walsh series W in terms of the A q constants [w] Aq for 1 ≤ q ≤ p. In particular, we show that, exactly as for the Hilbert transform, C L p (w) is bounded linearly by [w] Aq for 1 ≤ q < p. We also obtain L p (w) bounds in terms of [w] Ap , whose sharpness is related to certain conjectures (for instance, of Konyagin [27]) on pointwise convergence of Fourier series for functions near L 1 .
We prove a weak-L p bound for the Walsh-Carleson operator for p near 1, improving on a theorem of... more We prove a weak-L p bound for the Walsh-Carleson operator for p near 1, improving on a theorem of Sjölin . We relate our result to the conjectures that the Walsh-Fourier and Fourier series of a function f ∈ L log L( ) converge for almost every x ∈ .
Let Ω ⊂ R 2 be a bounded, simply connected domain with boundary ∂Ω of class C 1,1 except at finit... more Let Ω ⊂ R 2 be a bounded, simply connected domain with boundary ∂Ω of class C 1,1 except at finitely many points S j where ∂Ω is locally a corner of aperture α j ≤ π 2 . Improving on results of Grisvard , we show that the solution G Ω f to the Dirichlet problem on Ω
Let K be a Calderon-Zygmund convolution kernel on R. We discuss the L p -boundedness of the maxim... more Let K be a Calderon-Zygmund convolution kernel on R. We discuss the L p -boundedness of the maximal directional singular integral
As a model problem for the barotropic mode of the primitive equations of the oceans and atmospher... more As a model problem for the barotropic mode of the primitive equations of the oceans and atmosphere, we consider the Euler system on a bounded convex planar domain Ω, endowed with non-penetrating boundary conditions. For 4 3 ≤ p ≤ 2, and initial and forcing data with L p (Ω) vorticity we show the existence of a weak solution, enriching and extending the results of Taylor .
We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Os... more We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Oscillon equation
This paper is concerned with the integrodifferential equation $$\partial_t u-\Delta u -\int_0^\in... more This paper is concerned with the integrodifferential equation $$\partial_t u-\Delta u -\int_0^\infty \kappa(s)\Delta u(t-s)\,\d s + \varphi(u)=f$$ arising in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in presence of a nonlinearity $\varphi$ of critical growth. Rephrasing the equation within the history space framework, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related solution semigroup, acting both on the basic weak-energy space and on a more regular phase space.
Russian Journal of Mathematical Physics, Jan 1, 2009
We consider the singular limit of the semilinear strongly damped wave equation with memory in the... more We consider the singular limit of the semilinear strongly damped wave equation with memory in the presence of a critical nonlinearity, as the memory kernel converges to the Dirac mass at zero. We prove the existence of a robust family of regular exponential attractors in the weak energy space.
We establish a uniform domination of the family of trilinear multiplier forms with singularity o... more We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one-dimensional subspace by positive sparse forms involving L^p-averages. This class includes the adjoint forms to the bilinear Hilbert transforms. Our result strengthens the L^p-boundedness proved by Muscalu, Tao and Thiele and entails as a corollary a rich multilinear weighted theory. In particular, we obtain weighted boundedness of the bilinear Hilbert transform when the weights belong to the classes A_p intersection RH 2. Our proof relies on a stopping time construction based on newly developed localized outer-L p embedding theorems for the wave packet transform.
A semilinear strongly damped wave equation with memory is considered in the past history framewor... more A semilinear strongly damped wave equation with memory is considered in the past history framework. The existence of global attractors of optimal regularity is established, both for critical and supercritical nonlinearities, under a necessary and sufficient condition on the memory kernel.
We develop a Banach-valued version of the outer Lp space theory of Do and Thiele [7], relying on ... more We develop a Banach-valued version of the outer Lp space theory of Do and Thiele [7], relying on suitable randomized analogues of the conical square func- tion and nontangential maximal function. Within this framework, we prove an ab- stract Coifman-Meyer type, operator-valued multilinear multiplier theorem for suit- able tuples of UMD spaces. A concrete case of our theorem is a multilinear gener- alization of Weis’ operator-valued Ho ̈rmander-Mihlin linear multiplier theorem [44]. Furthermore, we derive from our abstract result a wide range of mixed Lp-norm es- timates for multi-parameter multilinear multiplier operators, as well as for the more singular tensor products of a one-parameter Coifman-Meyer multiplier with a bilin- ear Hilbert transform. These respectively extend the results of Muscalu et. al. [35] and of Silva [42]. In the same setting, we derive several operator-valued T(1)-type theorems both in one parameter, and of multi-parameter, mixed-norm type. A distin- guishing feature of our T(1) theorems is that the usual explicit assumptions on the distributional kernel of T are replaced with testing-type conditions.
We study the behavior of the bilinear Hilbert transform BHT at the boundary of the known boundedn... more We study the behavior of the bilinear Hilbert transform BHT at the boundary of the known boundedness region . A sample of our results is the estimate
We prove L p (w) bounds for the Carleson operator C, its lacunary version C lac , and its analogu... more We prove L p (w) bounds for the Carleson operator C, its lacunary version C lac , and its analogue for the Walsh series W in terms of the A q constants [w] Aq for 1 ≤ q ≤ p. In particular, we show that, exactly as for the Hilbert transform, C L p (w) is bounded linearly by [w] Aq for 1 ≤ q < p. We also obtain L p (w) bounds in terms of [w] Ap , whose sharpness is related to certain conjectures (for instance, of Konyagin [27]) on pointwise convergence of Fourier series for functions near L 1 .
We prove a weak-L p bound for the Walsh-Carleson operator for p near 1, improving on a theorem of... more We prove a weak-L p bound for the Walsh-Carleson operator for p near 1, improving on a theorem of Sjölin . We relate our result to the conjectures that the Walsh-Fourier and Fourier series of a function f ∈ L log L( ) converge for almost every x ∈ .
Let Ω ⊂ R 2 be a bounded, simply connected domain with boundary ∂Ω of class C 1,1 except at finit... more Let Ω ⊂ R 2 be a bounded, simply connected domain with boundary ∂Ω of class C 1,1 except at finitely many points S j where ∂Ω is locally a corner of aperture α j ≤ π 2 . Improving on results of Grisvard , we show that the solution G Ω f to the Dirichlet problem on Ω
Let K be a Calderon-Zygmund convolution kernel on R. We discuss the L p -boundedness of the maxim... more Let K be a Calderon-Zygmund convolution kernel on R. We discuss the L p -boundedness of the maximal directional singular integral
As a model problem for the barotropic mode of the primitive equations of the oceans and atmospher... more As a model problem for the barotropic mode of the primitive equations of the oceans and atmosphere, we consider the Euler system on a bounded convex planar domain Ω, endowed with non-penetrating boundary conditions. For 4 3 ≤ p ≤ 2, and initial and forcing data with L p (Ω) vorticity we show the existence of a weak solution, enriching and extending the results of Taylor .
We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Os... more We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Oscillon equation
This paper is concerned with the integrodifferential equation $$\partial_t u-\Delta u -\int_0^\in... more This paper is concerned with the integrodifferential equation $$\partial_t u-\Delta u -\int_0^\infty \kappa(s)\Delta u(t-s)\,\d s + \varphi(u)=f$$ arising in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in presence of a nonlinearity $\varphi$ of critical growth. Rephrasing the equation within the history space framework, we prove the existence of global and exponential attractors of optimal regularity and finite fractal dimension for the related solution semigroup, acting both on the basic weak-energy space and on a more regular phase space.
Russian Journal of Mathematical Physics, Jan 1, 2009
We consider the singular limit of the semilinear strongly damped wave equation with memory in the... more We consider the singular limit of the semilinear strongly damped wave equation with memory in the presence of a critical nonlinearity, as the memory kernel converges to the Dirac mass at zero. We prove the existence of a robust family of regular exponential attractors in the weak energy space.
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