This book collects selected peer reviewed papers on the topics of Nonlinear Analysis, Functional Analysis, (Korovkin-Type) Approximation Theory, and Partial Differential Equations. The aim of the volume is, in fact, to promote the connection among those different fields in Mathematical Analysis. The book celebrates Francesco Altomare, on the occasion of his 70th anniversary.

Heat Kernels and Spectral Theory investigates the theory of second-order elliptic operators.

We introduce conditions on the symmetric and skew-symmetric parts of timedependent, local, regular forms that imply a parabolic Harnack inequality for appropriate weak solutions of the associated heat equation, under natural assumptions on the underlying space. In particular, these local weak solutions are locally bounded and Holder continuous. Precise two-sided heat kernel estimates are deo rived from this parabolic Harnack inequality. For Dirichlet forms satisfying our conditions we prove a scale-invariant boundary Harnack principle in inner uniform domains. Inner uniformity is a condition on the boundary of the domain that is described solely in terms of the intrinsic length metric of the domain. In addition, we show that the Martin boundary of an inner uniform domain is homeomorphic to the boundary of the domain with respect to its completion in the inner distance. The main result of this work are two-sided Gaussian bounds for Dirichlet heat kernels corresponding to (non- )symmetric, local, regular Dirichlet forms. These bounds hold in domains that satisfy the inner uniformity condition. The proof uses the parabolic Harnack inequality and the boundary Harnack principle described above, as well as the Doob h-transform technique. For inner uniform Euclidean domains, our results apply to divergence form operators that are not necessarily symmetric, and complement earlier results by H. Aikawa, A. Ancona, P. Gyrya, L. Saloff-Coste and K.-T. Sturm.

This is the first comprehensive reference published on heat equations associated with non self-adjoint uniformly elliptic operators. The author provides introductory materials for those unfamiliar with the underlying mathematics and background needed to understand the properties of heat equations. He then treats Lp properties of solutions to a wide class of heat equations that have been developed over the last fifteen years. These primarily concern the interplay of heat equations in functional analysis, spectral theory and mathematical physics. This book addresses new developments and applications of Gaussian upper bounds to spectral theory. In particular, it shows how such bounds can be used in order to prove Lp estimates for heat, Schrödinger, and wave type equations. A significant part of the results have been proved during the last decade. The book will appeal to researchers in applied mathematics and functional analysis, and to graduate students who require an introductory text to sesquilinear form techniques, semigroups generated by second order elliptic operators in divergence form, heat kernel bounds, and their applications. It will also be of value to mathematical physicists. The author supplies readers with several references for the few standard results that are stated without proofs.

In the present book the conditions are studied for the semi-boundedness of partial differential operators which is interpreted in different ways. Nowadays one knows rather much about L2-semibounded differential and pseudo-differential operators, although their complete characterization in analytic terms causes difficulties even for rather simple operators. Until recently almost nothing was known about analytic characterizations of semi-boundedness for differential operators in other Hilbert function spaces and in Banach function spaces. The goal of the present book is to partially fill this gap. Various types of semi-boundedness are considered and some relevant conditions which are either necessary and sufficient or best possible in a certain sense are given. Most of the results reported in this book are due to the authors.