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In this short note, we prove Strichartz estimates for Schrödinger operators with slowly decaying singular potentials in dimension two. This is a generalization of the recent results by Mizutani, which are stated for dimension greater than two. The main ingredient of the proof is a variant of Kato’s smoothing estimate with a singular weight.

Here we discuss a new simplified proof of the essential self-adjointness for formally self-adjoint differential operators of real principal type, previously proved by Vasy (2020) and Nakamura-Taira (2021). For simplicity, here we discuss the second order cases, i.e., Klein-Gordon type operators only.

We study the essential self-adjointness for real principal type differential operators. Unlike the elliptic case, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients, and we prove the essential self-adjointness under the null non-trapping condition., Comment: 23pages

In this paper, we shall show that the limiting absorption principle for the wave operator on the asymptotically Minkowski spacetime. This problem was previously considered by [A. Vasy, J. Spect. Theory, 10,439-461 , (2020)]. Here, we employ a more transparent tool, the Mourre theory and removes an additional condition which is imposed in his paper. Moreover, we also prove that the anti-Feynman propagator defined by G\'erard and Wrochna coincides with the outgoing resolvent., Comment: Proof of Lemma 3.5 is corrected

We consider for discrete Schrodinger operators with dimension three or larger on the square lattice. We prove that the asymptotic behavior at infinity of resonances at elliptic thresholds is similar to those of continuous Schrodinger operators and that resonances are absent at hyperbolic thresholds. Moreover, we show the limiting absorption principle near the hyperbolic thresholds with the optimal weight.

In this short note, smoothness of the fundamental solution of Schrödinger equations on a complete manifold is studied. It is shown that (1) the fundamental solution is smooth under "mild" trapping conditions; (2) there is a Riemannian manifold which is equal to Euclidean space outside a compact set such that the fundamental solution is not smooth., 6 pages

In this note, uniform bounds of the Birman-Schwinger operators in the discrete setting are studied. For uniformly decaying potentials, we obtain the same bound as in the continuous setting. However, for non-uniformly decaying potential, our results are weaker than in the continuous setting. As an application, we obtain unitary equivalence between the discrete Laplacian and the weakly coupled systems.

In this note, we study a geometric property of asymptotically Minkowski spacetimes and an analytic property of the Klein-Gordon operator. Precisely, our first main results show that asymptotically Minkowski spacetimes are geodesically complete under a null non-trapping condition. Secondly, we prove that Sobolev index of a real principal type estimate used in the previous work is actually optimal.

In this paper, we study the mapping property form $L^p$ to $L^q$ of the resolvent of the Fourier multiplier operators and scattering theory of generalized Schr\"odinger operators. Though the first half of the subject is studied in [4], we extend their result to away from the duality line and we also study the H\"older continuity of the resolvent., Comment: 32pages

Familial combined hyperlipidemia (FCHL) is characterized by elevated levels of serum total cholesterol (TC), triglyceride (TG), or both. The increased incidence of coronary artery diseases (CAD) in the patients with FCHL is believed to be caused by circulating atherogenic lipoproteins associated with the complex phenotype. Recent establishment of sensitive detection system for malondialdehyde-modified (MDA)-LDL, which is one of oxidized lipoproteins, showed its increased circulating level in the patients with CAD. In order to know the atherogenic lipoproteins resulted from the dyslipidemia observed in FCHL, we measured the serum MDA-LDL level in the patients. The circulating MDA-LDL level and the ratio of MDA-LDL and LDL-C in FCHL were significantly higher (P

Patients with heterozygous familial hypercholesterolemia (FH) were examined with B-mode ultrasound in order to determine intima-media thickness (IMT) in the common carotid artery, and to uncover potential risk factors responsible for the development of IMT. Ninety seven FH subjects and 132 non FH type IIa hyperlipidemic subjects were involved in the present study. Age was found to correlate positively with IMT in both FH and non FH groups. FH individuals showed a higher IMT, along with elevated low density lipoprotein (LDL) cholesterol levels, compared with age-matched non FH individuals. To clarify potential factors contributing to the formation and development of carotid atherosclerosis, we divided the FH subjects into two subgroups, namely FH with high IMT group (HIG), and those with low IMT group (LIG). We investigated those two subgroups on the presence of angiographically documented coronary heart disease (CHD), of family history of CHD and of ‘midband lipoproteins’ by polyacrylamide gel electrophoresis (PAGE) analysis, by matching for age and LDL-cholesterol (LDL-C) level. Fifty percent of FH men in HIG was found to have CHD, whereas only 14% of those in LIG had CHD (P

In a previous study it was shown that postprandial lipid metabolism is delayed in individuals with intra-abdominal visceral fat accumulation. Population studies have shown that as compared with individuals with apolipoprotein (apo) E3/3, those with phenotype apo E3/4 phenotype have higher plasma and low density lipoprotein (LDL)-cholesterol (C) concentration and increased susceptibility to coronary heart disease. The aim of the present study is to determine how apo E4 affects postprandial lipid metabolism by comparing individuals with apo E3/4 to those with apo E3/3 phenotype matched for abdominal visceral fat. Sixty-two Japanese subjects (41 male, 21 female) [average age 48+/-14 years; mean body mass index (BMI) 25+/-5.6 kg/m2] were recruited for this study. The subjects were divided into two groups: those with apo E3/3 (n=43) and those with apo E3/4 phenotype (n=19), as determined by isoelectric focusing (IEF). Visceral fat accumulation was analyzed as area of fat deposition by computerized tomography at the umbilicus level. After a 12-h overnight fasting, an oral vitamin A and a fatty meal were administered to these subjects. The plasma triglyceride (TG) increased significantly hours after fat loading in both groups but the levels of TG were significantly higher in apo E3/4 than in apo E3/3 phenotype at 2, 4 and 6 h after fat loading. Plasma retinyl palmitate (RP) levels were also significantly higher in individuals with apo E3/4 than in those with apo E3/3 phenotype at 2, 4 and 6 h after fat loading. This investigation was then conducted in both genders separately, and found that these associations were statistically significant in men. Furthermore, after matching men for fasting TG levels, these associations did not persist for plasma TG levels at any time point, while plasma RP levels were still significantly higher in apo E3/4 group at 2 and 6 h after fat loading. These results indicate that in Japanese population especially for men apo E phenotype E3/4 is associated with an impaired postprandial TG-rich lipoprotein metabolism relative to apo E3/3 phenotype when matched for intra-abdominal visceral fat accumulation, which has a substantial effect on the metabolism of plasma TG-rich lipoproteins.

The subject was a 57-year-old Japanese woman with a body mass index of 21.2 kgm(-2). Her serum total cholesterol (TC), triglycerides (TG) and HDL-cholesterol levels were 7.11 mmoll(-1), 0.53 mmoll(-1) and 2.05 mmoll(-1), respectively. She had a marked increase of serum apolipoprotein (Apo) E concentration of 25 mgdl(-1) with normal concentrations of serum Apo A-I, A-II, B, C-II and C-III. Polymerase chain reaction-restriction fragments length polymorphism analysis of the cholesteryl ester transfer protein (CETP) gene from this subject revealed the heterozygous nucleotide change causing a Asp442 to Gly substitution (D442G) in the CETP protein. For comparison, 11 unrelated female subjects with this mutation (age, 57+/-5.1 years; BMI, 22+/-1.5 kgm(-2); TC, 7.23+/-1.16 mmoll(-1); TG, 1.44+/-0.80 mmoll(-1); HDL-C, 2.47+/-0.53 mmoll(-1)) were found to have a serum Apo E concentration of 7+/-1.5 mgdl(-1), about a third of the patient's concentration. The lipoprotein profile of the proband's serum analyzed by disk polyacrylamide gel electrophoresis showed a trace amount of VLDL. A vitamin A fat-loading test showed little increase in serum triglycerides and retinyl palmitate levels compared with control subjects at 2, 4 and 6 h after fat loading. Ultracentrifugation analysis of her serum revealed no detectable Apo E in the VLDL fraction but showed a large amount of Apo E in the HDL fraction, in contrast to a normal control, who had Apo E in the VLDL fraction as well as in the HDL fraction. Sequence analysis of the Apo E gene from the subject showed no nucleotide changes in exon 3 and exon 4, which code the mature Apo E protein, indicating there is no structural abnormality in the Apo E protein. Direct sequence analysis of the LDL receptor gene also did not show any nucleotide change. Based on these findings, it was hypothesized that the marked increase of Apo E in the patient's serum was caused by a decreased transfer of Apo E from HDL particles to TG-rich lipoproteins or impaired uptake of Apo E-containing HDL by LDL receptor or remnant receptor, due presumably to a dysfunction of these receptors in the patient.

Postprandial hyperlipidemia is frequently accompanied with intra-abdominal visceral accumulation in human subjects. We have found that the decreased lipoprotein lipase (LPL) mass and activity is negatively associated with the amount of visceral fat accumulation. Here, we studied the postprandial hyperlipidemia using the OLETF rat, a model with visceral obesity, in order to clarify the molecular mechanism causing postprandial hyperlipidemia accompanied with visceral obesity. At the same age of 32 weeks, the OLETF rats showed obviously higher plasma leptin, total cholesterol, triglyceride, and HDL-cholesterol levels than the control LETO rats, although the plasma glucose level was not significantly different. Fat-loading test revealed the delayed metabolism of exogenous fat in the OLETF rats compared to the LETO rats, similar to human subjects with visceral obesity. In the obese rats, plasma levels of LPL mass and activities were 60 and 49% of control rats. The expression of LPL gene was decreased in subcutaneous adipose tissues and skeletal muscle of OLETF rats to 40 and 52% compared to those of LETO rats. In OLETF rats, plasma tumor necrosis factor-alpha (TNF-alpha) and insulin levels were increased to 2.0- and 2.3-folds compared to those in control rats. Furthermore, plasma insulin and TNF-alpha levels in OLETF rats were negatively correlated with the expression levels of LPL gene in subcutaneous fat and muscle. These results indicate that decreased LPL mass and activity in the animal model with visceral obesity is possibly caused by decreased expression of LPL gene in tissues mediated by the increased levels of insulin and TNF-alpha. The different expression of LPL gene in tissues associated with the increased levels of insulin and TNF-alpha possibly elucidate the underlying mechanisms involving the postprandial hyperlipidemia observed in visceral obesity.

A 61-year-old Japanese woman with heterozygous familial hypercholesterolemia (FH), type 2 diabetes mellitus and coronary artery disease underwent coronary artery bypass grafting (CABG) utilizing a saphenous vein graft at the age of 46, in June 1984, 6 months before low density lipoprotein (LDL) apheresis was started. She had received LDL apheresis every two weeks, along with combined drug treatment since the age of 47 (December 1984). She had bilateral xanthelasma and Achilles tendon xanthomas. Her fasting baseline serum total cholesterol and triglyceride level were 464 mg/dl and 57 mg/dl, respectively at the age of 47 when she visited our hospital for the first time. Analysis of the genomic DNA from the patient revealed heterozygous amino acid substitution of Leu for Pro664 in the LDL receptor gene. She was diagnosed as type 2 diabetes mellitus at the age of 53. Combined treatment in the steady state yielded a pretreatment LDL cholesterol level of 230+/-14 mg/dl and a posttreatment level of 57+/-7.6. All grafts were widely patent after as long as 14 years since CABG, suggesting that LDL apheresis combined with drug therapy is highly effective in preventing the occlusion of bypass grafts in a patient with heterozygous FH and type 2 diabetes mellitus.

A new heterozygous lipoprotein lipase gene defect has been identified in a type I hyperlipidemic patient at the position of notable amino acid Asn 291. The patient is a 33-year-old male. His body mass index (BMI) was 18.5 kg/m2. The total cholesterol (TC), triglycerides (TG) and high density lipoprotein-cholesterol (HDL-C) concentration from his fasting plasma were 4.8, 11.9 and 0.4 mmol/l, respectively. The lipoprotein lipase (LPL) activity and mass in the postheparin plasma (PHP) from the patient were 0.58 mmol/ml/h (normal range: 7.7±2.6) and 244 ng/ml (normal range: 192±30), respectively. The hepatic lipase activity of the PHP from the patient was 10.6 mmol/ml/h (normal range: 9.9±3.6). DNA analysis of the LPL gene revealed that this patient had a heterozygous one nucleotide deletion of A coding Asn 291, resulting in a premature termination of the LPL protein at amino acid residue 303. The other abnormality in the LPL gene of the proband was an amino acid residue 194 defect (Ile194→Thr), which is known to cause a defective enzyme. A medium-chain triglyceride (MCT) loading test was conducted to find how this triglyceride affects plasma lipoprotein metabolism in this patient in a short term ( Fig. 3 ). The plasma total cholesterol (TC) or high density lipoprotein (HDL)-C levels did not change significantly after oral administration of a fatty meal containing long chain triglycerides (LCT) or MCT. The plasma TG level, on the other hand, increased from 11.9 to 19.2 mmol/l (+61%) at 6 h after loading a fatty meal containing LCT, whereas the plasma TG levels tended to even decrease at 6 h after oral administration of an MCT, tricaprin (from 11.6 to 10.5 mmol/l (−9.4%)). These results suggest that MCT, as opposed to LCT, is useful for treatment of type I hyperlipidemia with a novel mutation at the notable amino acid Asn 291 of the LPL gene.

Since the molecular identification of the low density lipoprotein receptor (LDLR), an ever increasing number of related proteins have been discovered. These receptors belonging to the LDLR family are thought to play key roles in lipoprotein metabolism in a variety of tissues, including the arterial wall. We have discovered that the expression of a 250-kDa mosaic LDLR-related protein, which we termed LR11 for the presence of 11 LDLR ligand-binding repeats, is markedly induced in smooth muscle cells in the hyperplastic intima of animal models used for the study of atherosclerosis. Here, we demonstrate that the human LR11, when overexpressed in hamster cells, binds and internalizes 39-kDa receptor-associated protein (RAP), an in vitro ligand for all receptors belonging to the LDLR family. Furthermore, LR11 binds the apolipoprotein E (apoE)-rich lipoproteins, beta-very low density lipoproteins (VLDLs), with a high affinity similar to that of other members, such as the LDLR and VLDL receptor. RAP and beta-VLDL compete with each other; however, other serum lipoproteins are not able to inhibit their binding. LR11 shows specific binding of apoE-enriched HDL prepared from human cerebrospinal fluid as well as of beta-VLDL, suggesting that the apoE content of lipoproteins is most likely important for mediating the high-affinity binding to the receptor. LR11-overexpressing cells are able to internalize and degrade the bound beta-VLDL; these cells also show increased accumulation of cholesteryl esters when incubated with beta-VLDL. Incubation for 48 hours with beta-VLDL of LR11-overexpressing cells, but not of control cells, promotes the appearance of numerous intracellular lipid droplets. Taken together, LR11, a mosaic LDLR family member whose expression in smooth muscle cells is markedly induced in atheroma, has all the properties of a receptor for the endocytosis of lipoproteins, particularly for the incorporation of apoE-rich lipoproteins.