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For a general measure space $(Omega,mu)$, it is shown that for every band $M$ in $L_p(mu)$ there exists a decomposition $mu=mu+mu^{primeprime}$ such that $M=L_p(mu)={fin L_p(mu);f=0 mu^{primeprime}text{-a.e.}}$. The theory is illustrated by an example, with an application to absorption semigroups.
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We study the dynamics of the group of isometries of $L_p$-spaces. In particular, we study the canonical actions of these groups on the space of $delta$-isometric embeddings of finite dimensional subspaces of $L_p(0,1)$ into itself, and we show that for $p eq 4,6,8,ldots$ they are $varepsilon$-transitive provided that $delta$ is small enough. We achieve this by extending the classical equimeasurability principle of Plotkin and Rudin. We define the central notion of a Fraisse Banach space which underlies these results and of which the known separable examples are the spaces $L_p(0,1)$, $p eq 4,6,8,ldots$ and the Gurarij space. We also give a proof of the Ramsey property of the classes ${ell_p^n}_n$, $p eq 2,infty$, viewing it as a multidimensional Borsuk-Ulam statement. We relate this to an arithmetic version of the Dual Ramsey Theorem of Graham and Rothschild as well as to the notion of a spreading vector of Matouv{s}ek and R{o}dl. Finally, we give a version of the Kechris-Pestov-Todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space $X$ with a Ramsey property of the collection of finite dimensional subspaces of $X$.
We show that there are $2^{2^{aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1))$, $1<p ot= 2<infty$. This solves a problem in A. Pietschs 1978 book Operator Ideals. The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non Hilbertian. We give a criterion for a space with an unconditional basis to have $2^{2^{aleph_0}}$ closed ideals in terms of the existence of a single operator on the space with some special asymptotic properties. We then show that for $1<q<2$ the space ${frak X}_q$ of Rosenthal, which is isomorphic to a complemented subspace of $L_q(0,1)$, admits such an operator.
The aim of this paper is to develop the $L_p$ John ellipsoid for the geometry of log-concave functions. Using the results of the $L_p$ Minkowski theory for log-concave function established in cite{fan-xin-ye-geo2020}, we characterize the $L_p$ John ellipsoid for log-concave function, and establish some inequalities of the $L_p$ John ellipsoid for log-concave function. Finally, the analog of Balls volume ratio inequality for the $L_p$ John ellipsoid of log-concave function is established.
In this paper, we investigate the wavelet coefficients for function spaces $mathcal{A}_k^p={f: |(i omega)^khat{f}(omega)|_pleq 1}, kin N, pin(1,infty)$ using an important quantity $C_{k,p}(psi)$. In particular, Bernstein type inequalities associated with wavelets are established. We obtained a sharp inequality of Bernstein type for splines, which induces a lower bound for the quantity $C_{k,p}(psi)$ with $psi$ being the semiorthogonal spline wavelets. We also study the asymptotic behavior of wavelet coefficients for both the family of Daubechies orthonormal wavelets and the family of semiorthogonal spline wavelets. Comparison of these two families is done by using the quantity $C_{k,p}(psi)$.
The classical Banach space $L_1(L_p)$ consists of measurable scalar functions $f$ on the unit square for which $$|f| = int_0^1Big(int_0^1 |f(x,y)|^p dyBig)^{1/p}dx < infty.$$ We show that $L_1(L_p)$ $(1 < p < infty)$ is primary, meaning that, whenever $L_1(L_p) = Eoplus F$ then either $E$ or $F$ is isomorphic to $L_1(L_p)$. More generally we show that $L_1(X)$ is primary, for a large class of rearrangement invariant Banach function spaces.
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