On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

Direct approach to <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates in homogenization theory

We derive interior L ^p -estimates for solutions of linear elliptic systems with oscillatory coefficients. The estimates are independent of ε, the small length scale of the rapid oscillations. So far, such results are based on potential theory and restricted to periodic coefficients. Our approach relies on BMO-estimates and an interpolation argument, gradients are treated with the help of finite differences. This allows to treat coefficients that depend on a fast and a slow variable. The estimates imply an L ^p -corrector result for approximate solutions.      

Rationality of the vertex operator algebra <Emphasis Type="Italic">V</Emphasis><Subscript><Emphasis Type="Italic">L</Emphasis></Subscript>+ for a positive definite even lattice <Emphasis Type="Italic">L</Emphasis>

The lattice vertex operator algebra V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from –1-isometry of L. The fixed point set V_L+ of V_L for the automorphism is naturally a vertex operator algebra. We prove that any ₀-graded weak V_L⁺-module is completely reducible.Supported by JSPS Research Fellowships for Young Scientists.     

Selberg＇s Normal Density Theorem for Automorphic L-Functions for GLm

Let π be an irreducible unitary cuspidal representation of GLm（AQ） with m ≥ 2, and L（s, Tr） the L-function attached to π. Under the Generalized Riemann Hypothesis for L（s,π）, we estimate the normal density of primes in short intervals for the automorphic L-function L（s, π）. Our result generalizes the corresponding theorem of Selberg for the Riemann zeta-function.     

Sharp asymptotics of the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> approximation error for interpolation on block partitions

Adaptive approximation (or interpolation) takes into account local variations in the behavior of the given function, adjusts the approximant depending on it, and hence yields the smaller error of approximation. The question of constructing optimal approximating spline for each function proved to be very hard. In fact, no polynomial time algorithm of adaptive spline approximation can be designed and no exact formula for the optimal error of approximation can be given. Therefore, the next natural question would be to study the asymptotic behavior of the error and construct asymptotically optimal sequences of partitions. In this paper we provide sharp asymptotic estimates for the error of interpolation by splines on block partitions in \mathbbR^d{\mathbb{R}^d} . We consider various projection operators to define the interpolant and provide the analysis of the exact constant in the asymptotics as well as its explicit form in certain cases.     

Zero-density estimates for automorphic <Emphasis Type="Italic">L</Emphasis>-functions

In this paper we prove zero-density estimates of the large sieve type for the automorphic L-function L（s, f × χ）, where f is a holomorphic cusp form and χ（mod q） is a primitive character.     

Entropy conditions for <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">r</Emphasis></Subscript>-convergence of empirical processes

The law of large numbers (LLN) over classes of functions is a classical topic of empirical processes theory. The properties characterizing classes of functions on which the LLN holds uniformly (i.e. Glivenko–Cantelli classes) have been widely studied in the literature. An elegant sufficient condition for such a property is finiteness of the Koltchinskii–Pollard entropy integral, and other conditions have been formulated in terms of suitable combinatorial complexities (e.g. the Vapnik–Chervonenkis dimension). In this paper, we endow the class of functions with a probability measure and consider the LLN relative to the associated L _r metric. This framework extends the case of uniform convergence over , which is recovered when r goes to infinity. The main result is a L _r -LLN in terms of a suitable uniform entropy integral which generalizes the Koltchinskii–Pollard entropy integral.      

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Boundedness of general index transforms

We establish the boundedness properties in L _p for a class of integral transformations with respect to an index of hypergeometric functions. In particular, by using the Riesz-Thorin interpolation theorem, we get the corresponding results in L _p (R ₊), 1 p 2, for the Kontorovich-Lebedev, Mehler-Fock, and Olevskii index transforms. An inversion theorem is proved for a general index transformation. The case p=2 is known as the Plancherel-type theory for this class of transformations.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 127–147, January–March, 2005.     

On the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> affine isoperimetric inequalities

We obtain an isoperimetric inequality which estimate the affine invariant p-surface area measure on convex bodies. We also establish the reverse version of L _p -Petty projection inequality and an affine isoperimetric inequality of Γ_− p K.     

Interpolation of nonlinear operators in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We show that Peetre’s classical interpolation theorem in weighted L _p -spaces is carried over to some classes of nonlinear operators containing in particular the Lipschitz operators and operators close to them in the properties satisfying less restrictive conditions than Lipschitz in each of the spaces of a Banach pair.     

Point sets with low <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-discrepancy

In this paper we study the L _p -discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L _p -discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L _p -discrepancy (p an even integer) of order which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L ₂-discrepancy of digitally shifted Hammersley point sets which show that the value of the L ₂-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these. This work is supported by the Austrian Research Fund (FWF), Project P17022-N12 and Project S8305.     

Approximation of sine-shaped functions by constants in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, <Emphasis Type="Italic">p</Emphasis> < 1

We investigate the best approximations of sine-shaped functions by constants in the spaces L_p for p < 1. In particular, we find the best approximation of perfect Euler splines by constants in the spaces L_p for certain p(0,1).Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 745–762, June, 2004.     

Sharp Markov brothers type inequality in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> on a closed interval

A recent modification of a classic Landau-Lifshitz equation that includes the socalled spin-transfer torque is widely recognized in physics community as a model of magnetization dynamics in certain nanodevices. Motivated by some experimental evidence, we introduce a generalization of this model, coupled Landau-Lifshitz equations with spin-transfer torque terms, and analyze it from dynamical systems standpoint. An explicit stability criterion for the critical points in terms of all parameters of the system is derived and illustrated with stability diagrams. Our analysis provides certain guidelines for the design of magnetic nanodevices with optimized response to control parameters.     

Strictly singular operators in pairs of <Emphasis Type="Italic">L</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript> space

Let E and F be Banach spaces. A linear operator from E to F is said to be strictly singular if, for any subspace Q ? E, the restriction of A to Q is not an isomorphism. A compactness criterion for any strictly singular operator from L_p to L_q is found. There exists a strictly singular but not superstrictly singular operator on L_p, provided that p ≠ 2.     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

Ordinal indices of Small Subspaces of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We calculate the ordinal L _p index defined in [3] for Rosenthal’s space X _p , \({\ell_p}\) and \({\ell_2}\). We show that an infinite-dimensional subspace of L _p \({(2 < p < \infty)}\) non-isomorphic to \({\ell_2}\) embeds in \({\ell_p}\) if and only if its ordinal index is the minimal possible. We also give a sufficient condition for a \({\mathcal{L}_p}\) subspace of \({\ell_p \oplus \ell_2}\) to be isomorphic to X _p .     

Optimal estimates for the semigroup generated by the classical Volterra operator on <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

Optimal upper bounds are given for the norm of the semigroup (e ^−tV ) _t≥0, where V is the classical Volterra operator acting on L _p [0,1], 1≤p≤∞. In particular, for every p∈[1,∞] we prove that

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> discrepancy of generalized two-dimensional Hammersley point sets

We determine the L _p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L _p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L _p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L _p discrepancy of the generalized Hammersley point set is of best possible order. For the L ₂ discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

A note on the complexity of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> minimization

We discuss the L _p (0 ≤ p < 1) minimization problem arising from sparse solution construction and compressed sensing. For any fixed 0 < p < 1, we prove that finding the global minimal value of the problem is strongly NP-Hard, but computing a local minimizer of the problem can be done in polynomial time. We also develop an interior-point potential reduction algorithm with a provable complexity bound and demonstrate preliminary computational results of effectiveness of the algorithm.