The ℓs\ell^s-boundedness of a family of integral operators on UMD Banach function spaces

We prove the $\ell^s$-boundedness of a family of integral operators with an operator-valued kernel on UMD Banach function spaces. This generalizes and simplifies earlier work by Gallarati, Veraar and the author, where the $\ell^s$-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of $\ell^s$-boundedness as weighted boundedness by Rubio de Francia.Comment: 13 pages. Generalization of arXiv:1410.665

Maximal regularity for non-autonomous equations with measurable dependence on time

In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the $L^p$-boundedness of a class of vector-valued singular integrals which does not rely on H\"ormander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of $m$-th order elliptic operators $A$ with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an $L^p(L^q)$-theory for such equations for $p,q\in (1, \infty)$. In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.Comment: Application to a quasilinear equation added. Accepted for publication in Potential Analysi

Maximal regularity for parabolic equations with measurable dependence on time and applications

The subject of this thesis is the study of maximal Lp-regularity of the Cauchy problem u'(t)+A(t)u(t)=f(t), t∈ (0,T), u(0)=x. We assume (A(t))_{t∈ (0,T)} to be a family of closed operators on a Banach space X0, with constant domain D(A(t))=X1 for every t∈ (0,T). Maximal Lp-regularity means that for all f∈ Lp(0,T;X0), the solution of the above evolution problem  is such that u', Au are both in Lp(0,T;X0).   In the first part of the thesis, we introduce a new operator theoretic approach to maximal Lp-regularity in the case the dependence t→A(t) is just measurable. The abstract method is then applied to concrete parabolic PDEs: we consider equations and systems of elliptic differential operators of even order, with coefficients measurable in the time variable and continuous in the space variables, and we show that they have maximal Lp-regularity on Lq(\Rd), for every p,q∈(1,∞). These results gives an alternative approach to several PDE results in the literature, where only the cases p=q or q≤p were considered. As a further example, we apply  our abstract approach also to higher order differential operators with general boundary conditions, on the half space, under the same assumptions.The last part of this thesis is based on a different approach and it is devoted to the study of maximal Lp-regularity on Lq(\Rd+) of an elliptic differential operator of higher order with coefficients in the class of vanishing mean oscillation both in the time and the space variables, and general boundary conditions of Lopatinskii-Shapiro type.Analysi

On the ℓs-boundedness of a family of integral operators

In this paper we prove an ℓs-boundedness result for integral operators with operator-valued kernels. The proofs are based on extrapolation techniques with weights due to Rubio de Francia. The results will be applied by the first and third author in a subsequent paper where a new approach to maximal Lp-regularity for parabolic problems with time-dependent generator is developed.Analysi

Propranolol eye drops in patients with corneal neovascularization

Rationale: Studies performed in animal models of corneal neovascularization suggested the possible efficacy of a treatment with propranolol. Corneal neovascularization is one of the most feared complications of Stevens–Johnson syndrome that frequently involves ocular surface. We report the first 2 patients with severe ocular neo-vascularization treated with different degrees of success, with propranolol eye drops. Patient concerns: Two patients with corneal neovascularization complicating the Stevens–Johnson syndrome, not responsive to steroids and cyclosporine, were treated with propranolol eye drops. Diagnoses: Corneal neovascularization was detected by ophthalmoscopic evaluation. Interventions: Topical treatment with propranolol eye drops at different concentrations. Outcomes: Both patients reported dramatic subjective benefits (reduction of photophobia and discomfort) without adverse effects, and in the patient with a less advanced disease, an objective reduction of neovascularization and an improved visual acuity was observed. Lessons: This experience suggests that propranolol might be an inexpensive, safe and effective treatment in counteracting the progression of corneal neovascularization

The ℓ ^s-boundedness of a family of integral operators on UMD banach function spaces

We prove the ℓs-boundedness of a family of integral operators with an operator-valued kernel on UMD Banach function spaces. This generalizes and simplifies earlier work by Gallarati, Veraar and the author, where the ℓs-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of ℓs-boundedness as weighted boundedness by Rubio de Francia.Analysi