A maximal LpL_p-regularity theory to initial value problems with time measurable nonlocal operators generated by additive processes

Let $Z=(Z_t)_{t\geq0}$ be an additive process with a bounded triplet $(0,0,\Lambda_t)_{t\geq0}$. Then the infinitesimal generators of $Z$ is given by time dependent nonlocal operators as follows: \begin{align*} \mathcal{A}_Z(t)u(t,x) &=\lim_{h\downarrow0}\frac{\mathbb{E}[u(t,x+Z_{t+h}-Z_t)-u(t,x)]}{h}=\int_{\mathbb{R}^d}(u(t,x+y)-u(t,x)-y\cdot \nabla u(t,x)1_{|y|\leq1})\Lambda_t(dy). \end{align*} Suppose that L\'evy measures $\Lambda_t$ have a lower bound (Assumption 2.10) and satisfy a weak-scaling property (Assumption 2.11). We emphasize that there is no regularity condition on L\'evy measures $\Lambda_t$ and they do not have to be symmetric. In this paper, we establish the $L_p$-solvability to initial value problem (IVP) \begin{equation} \label{20.07.15.17.02} \frac{\partial u}{\partial t}(t,x)=\mathcal{A}_Z(t)u(t,x),\quad u(0,\cdot)=u_0,\quad (t,x)\in(0,T)\times\mathbb{R}^d, \end{equation} where $u_0$ is contained in a scaled Besov space $B_{p,q}^{s;\gamma-\frac{2}{q}}(\mathbb{R}^d)$ (see Definition 2.8) with a scaling function $s$, exponent $p \in (1,\infty)$, $q\in[1,\infty)$, and order $\gamma \in [0,\infty)$. We show that IVP is uniquely solvable and the solution $u$ obtains full-regularity gain from the diffusion generated by a stochastic process $Z$. In other words, there exists a unique solution $u$ to IVP in $L_q((0,T);H_p^{\mu;\gamma}(\mathbb{R}^d))$, where $H_p^{\mu;\gamma}(\mathbb{R}^d)$ is a generalized Bessel potential space (see Definition 2.3). Moreover, the solution $u$ satisfies $$\|u\|_{L_q((0,T);H_p^{\mu;\gamma}(\mathbb{R}^d))}\leq N(1+T^2)\|u_0\|_{B_{p,q}^{s;\gamma-\frac{2}{q}}(\mathbb{R}^d)},$$ where $N$ is independent of $u$, $u_0$, and $T$.Comment: 44 page

An existence and uniqueness result to evolution equations with sign-changing pseudo-differential operators and its applications to logarithmic Laplacian operators and second-order differential operators without ellipticity

We broaden the domain of the Fourier transform to contain all distributions without using the Paley-Wiener theorem and devise a new weak formulation built upon this extension. This formulation is applicable to evolution equations involving pseudo-differential operators, even when the signs of their symbols may vary over time. Notably, our main operator includes the logarithmic Laplacian operator $\log (-\Delta)$ and a second-order differential operator whose leading coefficients are not positive semi-definite.Comment: 49 page

Effects of emotional labor on musculoskeletal disorders among physical therapists in Seoul

Introduction: Health care workers, including physical therapists, have some of the most important roles in the health care system as shown during the COVID-19 pandemic. Physical therapists encounter emotionally and physically vulnerable patients, experience emotional labor, and are exposed to conditions that can lead to job stress and musculoskeletal disorders. We aimed to examine the relationship between physical therapists’ emotional labor and its effect on perceived job stress and risk of musculoskeletal disorders. Methods: We conducted a 30-day survey among 230 physical therapists working in various settings from October 2 to November 1, 2019. Questionnaires, including questions on musculoskeletal symptoms, perceived job stress, and emotional labor, were administered to the participants. Results: The physical therapist's surface behavior affected the body burden. Job burnout experienced by physical therapists had an effect on their interpersonal relationships. The physical therapist's emotional law affects the degree of compensation. Conclusion: To prevent the long-term consequences of work-related strain, physical therapists should receive support in terms of maintaining a healthy lifestyle and developing effective methods of communication with patients. Encouragement of activities for psychological rejuvenation with colleagues with whom they can share emotional difficulties is also desirable. It is also necessary to establish a communication channel that can directly convey the grievances of physical therapists to hospitals

A regularity theory for parabolic equations with anisotropic non-local operators in Lq(Lp)L_{q}(L_{p}) spaces

In this paper, we present an $L_q(L_p)$-regularity theory for parabolic equations of the form: $$\partial_t u(t,x)=\mathcal{L}^{\vec{a},\vec{b}}(t)u(t,x)+f(t,x),\quad u(0,x)=0.$$ Here, $\mathcal{L}^{\vec{a},\vec{b}}(t)$ represents anisotropic non-local operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: $$\mathcal{L}^{\vec{a},\vec{0}}(t)u(x)=\sum_{i=1}^{d} \int_{\mathbb{R}}\left( u(x^{1},\dots,x^{i-1},x^{i}+y^{i},x^{i+1},\dots,x^{d}) - u(x) \right) \frac{a_{i}(t,y^{i})}{|y^{i}|^{1+\alpha_{i}}} \mathrm{d}y^{i} .$$ To address the anisotropy of the operator, we employ a probabilistic representation of the solution and Calder\'on-Zygmund theory. As applications of our results, we demonstrate the solvability of elliptic equations with anisotropic non-local operators and parabolic equations with isotropic non-local operators

On the trace theorem to Volterra-type equations with local or non-local derivatives

This paper considers traces at the initial time for solutions of evolution equations with local or non-local derivatives in vector-valued $A_p$ weighted $L_p$ spaces. To achieve this, we begin by introducing a generalized real interpolation method. Within the framework of generalized interpolation theory, we make use of stochastic process theory and two-weight Hardy's inequality to derive our trace and extension theorems. Our results encompass findings applicable to time-fractional equations with broad temporal weight functions

Characterizations of weighted Besov and Triebel-Lizorkin spaces with variable smoothness

In this paper, we study different types of weighted Besov and Triebel-Lizorkin spaces with variable smoothness. The function spaces can be defined by means of the Littlewood-Paley theory in the field of Fourier analysis, while there are other norms arising in the theory of partial differential equations such as Sobolev-Slobodeckij spaces. It is known that two norms are equivalent when one considers constant regularity function spaces without weights. We show that the equivalence still holds for variable smoothness and weights, which is accomplished by making use of shifted maximal functions, Peetre's maximal functions, and the reverse H\"older inequality. Moreover, we obtain a weighted regularity estimate for time-fractional evolution equations and a generalized Sobolev embedding theorem without weights.Comment: 36 page

Cardiac interventions in patients with achondroplasia: a systematic review.

Patients with achondroplasia and other causes of dwarfism suffer from increased rates of cardiovascular disease relative to the remainder of the population. Few studies have examined these patients when undergoing cardiac surgery or percutaneous intervention. This systematic review examines the literature to determine outcomes following cardiac intervention in this unique population. An electronic search was performed in the English literature to identify all reports of achondroplasia, dwarfism, and cardiac intervention. Of the 5,274 articles identified, 14 articles with 14 cases met inclusion criteria. Patient-level data was extracted and analyzed. Median patient age was 55.5 [interquartile ranges (IQR), 43.8, 59.8] years, median height 102.0 [98.8, 112.5] cm, median BMI 32.1 [27.0, 45.9], and 57.1% (8/14) were male. Of these 14 patients, nine had the following documented skeletal abnormalities: 66.7% (6/9) had scoliosis, 66.7% (6/9) had kyphosis, 11.1% (1/9) had lordosis, 11.1% (1/9) pectus carinatum and 11.1% (1/9) spinal stenosis. Coronary artery disease was present in 53.8% (7/13), and 30.8% (4/13) patients previously suffered a myocardial infarction. Of the eight patients who underwent cardiac surgery, 37.5% (3/8) underwent multivessel coronary artery bypass grafting, 37.5% (3/8) underwent aortic valve replacement, 25.0% (2/8) underwent type A aortic dissection repair, and the remaining 12.5% (1/8) underwent pulmonary thromboendarterectomy. Six patients underwent percutaneous intervention. Median cardiopulmonary bypass time was 136.5 [110.0, 178.8] minutes. Median arterial cannula size was 20.0 [20.0, 24.0] Fr. Bicaval cannulation was performed in all cases describing cannulation strategy (5/5). Median superior vena cava cannula size was 28.0 [28.0, 28.0] Fr, and inferior vena cava cannula size was 28.0 [28.0, 28.0] Fr. No mortality was reported with a median follow up time of 6.0 [6.0, 10.5] months. In conclusion, Common cardiac procedures can be performed with reasonable safety in this patient population. Operative adjustments may need to be made with respect to equipment to accommodate patient-specific needs