Lp Estimates Research Articles

Intracellular ice formation in mouse zygotes and early morulae vs. cooling rate and temperature-experimental vs. theory

In this study, mature female mice of the ICR strain were induced to superovultate, mated, and collected at either zygote or early morula stages. Embryos suspended in 1 M ethylene glycol in PBS containing 10 mg/L Snomax for 15 min, then transferred in sample holder to Linkam cryostage, cooled to and seeded at 7 °C, and then observed and photographed while being cooled to −70 °C at 0.5–20 °C/min. Intracellular ice formation (IIF) was observed as abrupt ‘‘flashing’’. Two types of flashing or IIF were observed in this study. Extracellular freezing occurred at a mean of −7.7 °C. In morulae, about 25% turned dark within ±1 °C of extracellular ice formation (EIF). These we refer to as “high temperature’’ flashers. In zygotes, there were no high temperature flashers. All the zygotes flashed at temperatures well below the temperature for EIF. Presumably high temperature flashers were a consequence of membrane damage prior to EIF or damage from EIF. We shall not discuss them further. In the majority of cases, IIF occurred well below −7.7 °C; these we call ‘‘low temperature’’ flashers. None flashed with cooling rate (CR) of 0.5 °C/min in either zygotes or morulae. Nearly all flashed with CR of 4 °C/min or higher, but the distribution of temperatures is much broader with morulae than with zygotes. Also, the mean flashing temperature is much higher with morulae (−20.9 °C) than with zygotes (−40.3 °C). We computed the kinetics of water loss with respect to CR and temperature in both mouse zygotes and in morulae based on published estimates of Lp and it is Ea. The resulting dehydration curves combined with knowledge of the embryo nucleation temperature permits an estimate of the likelihood of IIF as a function of CR and subzero temperature. The agreement between these computed probabilities and the observed values are good.

Revisit lipoprotein(a): An independent risk factor for coronary artery disease severity in Indian population and its correlation with Gensini score

AimsLipoprotein(a) [Lp(a)] levels have shown wide ethnic variations. Further studies are needed, as there are sparse data in Indian population on mean Lp(a) levels and its link with clinical variables and severity of coronary artery disease (CAD). MethodsIn present study total 516 patients of suspected CAD undergoing coronary angiography were included after applying inclusion and exclusion criteria on the target population. Serum Lp(a) estimation was performed by immunoturbidimetric method. ResultsAmong patients with normal coronaries, 67.3% patients had lipoprotein(a) levels <25 and 32.7% patients had lipoprotein(a) levels ≥25. In patients where 1 vessel was involved, 66.7% patients had lipoprotein(a) levels <25 and 33.3% patients had lipoprotein(a) levels ≥25. In patients where 2 vessels were involved 56% patients had lipoprotein(a) levels <25 and 44% patients had lipoprotein(a) levels ≥25. In patients where 3 vessels were involved 27.6% patients had lipoprotein(a) levels <25 and 72.6% patients had lipoprotein(a) levels ≥25. These results were significant and the p-value <0.001. The Gensini score was 29, when patients had lipoprotein(a) levels <25, while the Gensini score was 61, when patients had lipoprotein(a) levels ≥25 (p-value <0.001). Mean number of diseased vessels, when lipoprotein(a) levels were <25 was 1, while the mean number of vessels were 3 when the lipoprotein(a) levels were ≥25 (p-value of <0.001). ConclusionLipoprotein(a) was found to be an important independent risk factor for CAD in Indian population and raised Lp(a) levels were also associated with increasing severity of the coronary vessel disease and multivessel CAD.

Global dynamics of the Boussinesq‐Burgers system with large initial data

In this paper, we study the global existence and asymptotic behavior of the Boussinesq‐Burgers system subject to the Dirichlet boundary conditions. Based on the Lp(p &gt; 2) estimates of the solution, which are different from the standard L2‐based energy methods, we show that the classical solutions exist globally and converge to their boundary data at an exponential decay rate as time goes to infinity for large initial data. Copyright © 2016 John Wiley &amp; Sons, Ltd.

On the Challenge of Interpreting Census Data: Insights from a Study of an Endangered Pinniped.

Population monitoring is vital for conservation and management. However, simple counts of animals can be misleading and this problem is exacerbated in seals (pinnipeds) where individuals spend much time foraging away from colonies. We analyzed a 13-year-series of census data of Galapagos sea lions (Zalophus wollebaeki) from the colony of Caamaño, an islet in the center of the Galapagos archipelago where a large proportion of animals was individually marked. Based on regular resighting efforts during the cold, reproductive (cold-R; August to January) and the warm, non-reproductive (warm-nR; February to May) season, we document changes in numbers for different sex and age classes. During the cold-R season the number of adults increased as the number of newborn pups increased. Numbers were larger in the morning and evening than around mid-day and not significantly influenced by tide levels. More adults frequented the colony during the warm-nR season than the cold-R season. Raw counts suggested a decline in numbers over the 13 years, but Lincoln-Petersen (LP-) estimates (assuming a closed population) did not support that conclusion. Raw counts and LP estimates were not significantly correlated, demonstrating the overwhelming importance of variability in attendance patterns of individuals. The probability of observing a given adult in the colony varied between 16% (mean for cold-R season) and 23% (warm-nR season) and may be much less for independent 2 to 4 year olds. Dependent juveniles (up to the age of about 2 years) are observed much more frequently ashore (35% during the cold-R and 50% during the warm-nR seasons). Simple counts underestimate real population size by a factor of 4–6 and may lead to erroneous conclusions about trends in population size.

Gaussian heat kernel bounds through elliptic Moser iteration

On a doubling metric measure space endowed with a “carré du champ”, we consider Lp estimates (Gp) of the gradient of the heat semigroup and scale-invariant Lp Poincaré inequalities (Pp). We show that the combination of (Gp) and (Pp) for p≥2 always implies two-sided Gaussian heat kernel bounds. The case p=2 is a famous theorem of Saloff-Coste, of which we give a shorter proof, without parabolic Moser iteration. We also give a more direct proof of the main result in [37]. This relies in particular on a new notion of Lp Hölder regularity for a semigroup and on a characterisation of (P2) in terms of harmonic functions.

Global existence and decay rate of the Boussinesq–Burgers system with large initial data

In this paper, we study the global existence and the long-time asymptotic behavior of classical solutions to the Cauchy problem for the Boussinesq–Burgers system with large initial data. More precisely, we first show that the classical solutions exist globally in time with large initial data by using the Lp (p>2) estimates other than the conventional L2 estimate. Then we prove that the global classical solutions converge to constant equilibrium states with an algebraic decay rate as time approaches infinity.

Regularity for weak solutions to nondiagonal quasilinear degenerate elliptic systems

The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to Hörmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We first prove Lp (p≥2) estimates for gradients of weak solutions by using a priori estimates and a known reverse Hölder inequality, and consider regularity to the corresponding nondiagonal homogeneous degenerate elliptic systems. Then we get higher Morrey and Campanato estimates for gradients of weak solutions to original systems and Hölder estimates for weak solutions.

Extrapolation and weighted norm inequalities between Lebesgue and Lipschitz spaces in the variable exponent context

We give extrapolation results starting from weighted inequalities between Lebesgue and Lipschitz spaces, given by(0.1)supB⁡‖wχB‖∞|B|1+δn∫B|f(x)−mB(f)|dx≤C‖gw‖s, where 1<β<∞, 0≤δ<1, δn=1β−1s, f and g are two measurable functions and w belongs to a suitable class of weights. From this hypothesis we obtain a large class of inequalities including weighted Lp–Lq estimates and weighted Lp-Lipschitz integral spaces, generalizing well known results for certain sublinear operator. From the same hypothesis (0.1) we obtain the corresponding results in the setting of variable exponent spaces. Particularly, we obtain estimates of the type Lp(⋅)-variable versions of Lipschitz integral spaces. We also prove a surprising weighted inequalities of the type Lp(⋅)–Lq(⋅). An important tool in order to get the main results is an improvement of an estimate due to Calderon and Scott in [1], which allows us to relate different integral Lipschitz spaces. Our results are new even in the classical context of constant exponents.

Characterizing best isotone approximations in [formula omitted] spaces, [formula omitted

We characterize best approximations in Lp(1≤p<∞) from a class of convex subcones of the convex cone of nonnegative functions and the convex cone of increasing functions; in particular, from the convex cones themselves. These subcones arise naturally in applications of constrained or shape-preserving approximation.

Conormal problem of higher-order parabolic systems with time irregular coefficients

The paper is a comprehensive study of Lp and Schauder estimates for higher-order divergence type parabolic systems with discontinuous coefficients on a half space and cylindrical domains with the conormal derivative boundary conditions. For the Lp estimates, we assume that the leading coefficients are only bounded and measurable in the t variable and have vanishing mean oscillations (VMOx) with respect to x. We also prove the Schauder estimates in two situations: the coefficients are Holder continuous only in the x variable; the coefficients are Holder continuous in the t variable as well on the lateral boundary.

Modulation space estimates for damped fractional wave equation

Instead of the Lp estimates, we study the modulation space estimates for the solution to the damped wave equation. Decay properties for both the linear and semilinear equations are obtained. The estimates in modulation space differ in many aspects from those in Lp space.

Robustness of one-dimensional viscous fluid motion under multidimensional perturbations

We adapt the relative energy functional associated to the compressible Navier–Stokes system to show stability of solutions emanating from 1-D initial data with respect to multidimensional N=2,3 perturbations. Besides the application of the relative energy inequality as a suitable “distance” between two solutions, refined regularity estimates in Lp based Sobolev spaces are used.

Sharp martingale inequalities and applications to Riesz transforms on manifolds, Lie groups and Gauss space

We prove new sharp Lp, logarithmic, and weak-type inequalities for martingales under the assumption of differential subordination. The Lp estimates are “Feynman–Kac” type versions of Burkholder's celebrated martingale transform inequalities. From the martingale Lp inequalities we obtain that Riesz transforms on manifolds of nonnegative Bakry–Emery Ricci curvature have exactly the same Lp bounds as those known for Riesz transforms in the flat case of Rd. From the martingale logarithmic and weak-type inequalities we obtain similar inequalities for Riesz transforms on compact Lie groups and spheres. Combining the estimates for spheres with Poincaré's limiting argument, we deduce the corresponding results for Riesz transforms associated with the Ornstein–Uhlenbeck semigroup, thus providing some extensions of P.A. Meyer's Lp inequalities.

Nonlinear tensor product approximation of functions

We are interested in approximation of a multivariate function f(x1,…,xd) by linear combinations of products u1(x1)⋯ud(xd) of univariate functions ui(xi), i=1,…,d. In the case d=2 it is the classical problem of bilinear approximation. In the case of approximation in the L2 space the bilinear approximation problem is closely related to the problem of singular value decomposition (also called Schmidt expansion) of the corresponding integral operator with the kernel f(x1,x2). There are known results on the rate of decay of errors of best bilinear approximation in Lp under different smoothness assumptions on f. The problem of multilinear approximation (nonlinear tensor product approximation) in the case d≥3 is more difficult and much less studied than the bilinear approximation problem. We will present results on best multilinear approximation in Lp under mixed smoothness assumption on f.

Generalized impedance boundary conditions for strongly absorbing obstacle: The full wave equation

This paper is devoted to the study of the generalized impedance boundary conditions (GIBCs) for a strongly absorbing obstacle in the time regime in two and three dimensions. The GIBCs in the time domain are heuristically derived from the corresponding conditions in the time harmonic regime. The latter is frequency-dependent except the one of order 0; hence the formers are non-local in time in general. The error estimates in the time regime can be derived from the ones in the time harmonic regime when the frequency dependence is well controlled. This idea is originally due to Nguyen and Vogelius [Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807] for the cloaking context. In this paper, we present the analysis to the GIBCs of orders 0 and 1. To implement the ideas in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807], we revise and extend the work of Haddar, Joly, and Nguyen, [Generalized impedance boundary condition for scattering by strongly absorbing obstacles: the scalar case, Math. Models Methods Appl. Sci.15 (2005) 1273–1300], where the GIBCs were investigated for a fixed frequency in three dimensions. Even though we heavily follow the strategy in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807], our analysis on the stability contains new ingredients and ideas. First, instead of considering the difference between solutions of the exact model and the approximate model, we consider the difference between their derivatives in time. This simple idea helps us to avoid the machinery used in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variables, SIAM J. Math. Anal.44 (2012) 769–807] concerning the integrability with respect to frequency in the low frequency regime. Second, in the high frequency regime, the Morawetz multiplier technique used in [H.-M. Nguyen and M. S. Vogelius, Approximate cloaking for the full wave equation via change of variable, SIAM J. Math. Anal.44 (2012) 769–807] does not fit directly in our setting. Our proof makes use of a result by Hörmander in [Lp estimates for (pluri-) subharmonic functions, Math. Scand.20 (1967) 65–78]. Another important part of the analysis in this paper is the well-posedness in the time domain for the approximate problems imposed with GIBCs on the boundary of the obstacle, which are non-local in time.

Lp approximation with rates by multivariate generalized discrete singular operators

Here we give the approximation properties with rates of multivariate generalized discrete versions of Picard, Gauss–Weierstrass, and Poisson–Cauchy singular operators over RN,N ≥ 1. We treat both the unitary and non-unitary cases of the operators above. We derive quantitatively Lp convergence of these operators to the unit operator by involving the Lp higher modulus of smoothness of an Lp function.

Nonlocal Navier–Stokes problems in abstract function spaces and applications

The existence, uniqueness and Lp estimates to solution of initial nonlocal boundary value problems for Navier–Stokes equation with operator coefficients are derived. These types of boundary conditions arise in control problems, especially in experimental studies of incompressible fluids.

L'équation de Poisson pour les formes différentielles sur un espace symétrique et ses applications

We prove Lp and weighted Lp estimates for the solutions α of the Poisson equation Δα=β on differential forms with data β in Lp or in weighted Lp, in a symmetric space M. We study carefully the range of p for which the de Rham–Hodge projection, a priori defined on L2, is bounded on Lp.

The [formula omitted] endpoint estimate for maximal singular integral operators

We prove in this paper the following estimate for the maximal operator T⁎ associated to the singular integral operator T:‖T⁎f‖L1,∞(w)≲1ϵ∫Rn|f(x)|ML(log⁡L)ϵ(w)(x)dx,w≥0,0<ϵ≤1. This follows from the sharp Lp estimate‖T⁎f‖Lp(w)≲p′(1δ)1/p′‖f‖Lp(ML(log⁡L)p−1+δ(w)),1<p<∞,w≥0,0<δ≤1. As a consequence we deduce that‖T⁎f‖L1,∞(w)≲[w]A1log⁡(e+[w]A∞)∫Rn|f|wdx, extending the endpoint results obtained in [23] and [15] to maximal singular integrals. Another consequence is a quantitative two weight bump estimate.

One modification of the martingale transform and its applications to paraproducts and stochastic integrals

In this paper we introduce a variant of Burkholder's martingale transform associated with two martingales with respect to different filtrations. Even though the classical martingale techniques cannot be applied, we show that the discussed transformation still satisfies some expected Lp estimates. Then we apply the obtained inequalities to general-dilation twisted paraproducts, particular instances of which have already appeared in the literature. As another application we construct stochastic integrals ∫0tHsd(XsYs) associated with certain continuous-time martingales (Xt)t≥0 and (Yt)t≥0. The process (XtYt)t≥0 is shown to be a “good integrator”, although it is not necessarily a semimartingale, or even adapted to any convenient filtration.