Search

The aim of this paper is to construct (explicit) heat kernels for some hybrid evolution equations which arise in physics, conformal geometry and subelliptic PDEs. Hybrid means that the relevant partial differential operator appears in the form ${\mathscr{L}}_{1} + {\mathscr{L}}_{2} - \partial _{t}$ L 1 + L 2 − ∂ t , but the variables cannot be decoupled. As a consequence, the relative heat kernel cannot be obtained as the product of the heat kernels of the operators ${\mathscr{L}}_{1} - \partial _{t}$ L 1 − ∂ t and ${\mathscr{L}}_{2} - \partial _{t}$ L 2 − ∂ t . Our approach is new and ultimately rests on the generalised Ornstein-Uhlenbeck operators in the opening of Hörmander’s 1967 groundbreaking paper on hypoellipticity.

In this note we review a recent result in [17] in collaboration with N. Garofalo, where we establish global versions of Hardy-Littlewood-Sobolev inequalities attached to hypoelliptic equations of Kolmogorov type. The relevant Sobolev spaces are defined through the fractional powers of the operator under consideration. We outline the main steps of the semigroup approach we adopt., Bruno Pini Mathematical Analysis Seminar, Vol 11, No 1 (2020): Something about nonlinear problems

In this note we review some recent results in [64, 95, 96] concerning necessary and sufficient conditions for the regularity of boundary points relatively to the Dirichlet problem for linear degenerate-parabolic operators with well-behaved fundamental solutions. The main focus is on Wiener-type criteria for a class of operators whose degeneracy is controlled by H¨ormander vector fields.

Adult intussusception is a rare cause of bowel obstruction with atypical presentation. It can be associated with primary or secondary intestinal tumors and, rarely, with lipomatous masses.We report the case of a 69-year old man presenting with a history of gastrointestinal bleeding and anemia. Upper and lower endoscopies were negative for bleeding. On abdominal contrast enhanced computerized tomography (CT) scan, a trans-omental hernia in the right lower abdominal quadrant was diagnosed with no active bleeding or evidence of tumor. On exploratory laparoscopy we detected an ileo-ileal intussusception caused by a submucosal mass in the distal ileum, which was reduced and we then performed a segmental resection of the involved small bowel tract. The patient fully recovered by postoperative day 3 when he was discharged home. Final pathology confirmed an ileal lipoma.Ileal intussusception caused by lipoma is a rare condition, which can be diagnosed with endoscopy, barium enema, and abdominal ultrasound or CT scanning, but preoperative diagnosis may be difficult. The treatment of choice is the reduction of the intussusception and the resection (laparoscopic or open) of the involved tract.Intussusception, Lipoma, Ileum, Laparoscopy, Bleeding.L’intussuscezione intestinale dell’adulto è una rara condizione di occlusione intestinale che spesso ha una presentazione atipica. Può essere associato a tumori intestinali primitivi o secondari e, raramente, a masse lipomatose. Anche se l’esatto meccanismo per cui avviene l’intussuscezione non è conosciuto, si ritiene che sia associato a movimenti anomali della parete intestinale, che causerebbero l’ostruzione.Riportiamo il caso clinico di un uomo di 69 anni che si presenta alla nostra osservazione con una storia di sanguinamento gastrointestinale e anemizzazione. Gli esami endoscopici (EGDS e coloscopia) erano entrambi negativi per sanguinamento. La Tc addome con mezzo di contrasto evidenziava un’ernia transomentale nel quadrante addominale inferiore destro senza segni di sanguinamento attivo o evidenza di tumore. La successiva laparoscopia esplorativa ha rilevato una intussuscezione ileo-ileale causata da una massa sottomucosa nell’ileo distale; pertanto è stata eseguita la riduzione laparoscopica e resezione segmentale del tratto ileale coinvolto. Il paziente è stato dimesso in III giornata post-operatoria in buone condizioni cliniche generali. L’esame istologico ha confermato l’origine lipomatosa della lesione. CONCLUSIONE: l’intussuscezione ileale causata da lipoma è una condizione rara, che può essere diagnosticata con metodiche di imaging, ma spesso la diagnosi preoperatoria è difficile. Il trattamento di scelta è la riduzione chirurgica dell’intussuscezione e la resezione (laparoscopica o laparotomica) del tratto in questione.

We investigate the so-called Hopf lemma for certain degenerate-elliptic equations at characteristic boundary points of bounded open sets. For such equations, the validity of the Hopf lemma is related to the fact that the boundary of the open set reflects the underlying geometry of the specific operator. We present here some recent results obtained in [21] in collaboration with V. Martino. Our main focus is on conditions on the boundary which are stable by changing our operators in some particular classes, for example in the class of horizontally elliptic operators in non-divergence form. We also study what happens to these conditions for degenerate operators with first order terms., Bruno Pini Mathematical Analysis Seminar, Seminars 2015

We performed a multicenter prospective study on the incidence and prognosis of Guillain-Barre syndrome (GBS) in the Emilia-Romagna region (3,909,512 inhabitants), Italy, over a 2-year period (1992-1993). The case finding method was based on a surveillance system including neurologic departments, all private and public general hospitals, and all practicing neurologists. We also reviewed all the ICD codes of hospital discharges. Ninety-four patients with GBS were identified, giving a mean incidence rate of 1.20 per 100,000 per year. Men/women ratio was 1.94. Incidence rate increased with age for both sexes. There were no significant incidence variations among seasons or months. Antecedent infections were reported in 59% of the patients. There was an overall good prognosis. After 6 months, 64% of patients had recovered and only 4% was still severely disabled. Acute mortality was 3%. Older age was the strongest predictor of poor outcome. NEUROLOGY 1997;48: 214-221

Peripheral neuropathy is frequently reported in mixed cryoglobulinemia. As a close relationship has been established between mixed cryoglobulinemia and hepatitis C virus (HCV), the clinical, neurophysiological, and serologic findings of five patients affected by neuropathy, cryoglobulinemia and HCV infection were investigated. HCV infection was ascertained by the presence in the serum of anti-HCV antibodies detected by enzyme-linked immunosorbent assay(ELISA), and of HCV RNA assessed by polymerase chain reaction (PCR). Initial symptoms included paresthesias and painful dysesthesias in the legs. Later on, the patients developed a mainly asymmetric axonal polyneuropathy or a multifocal neuropathy associated with signs of systemic vasculitis. In this case series we report the short-term prognosis, as well as the response to interferon (IFN) alpha and conventional treatment. The presence of HCV RNA supports the hypothesis that a persistent HCV infection might be involved in the pathogenesis of mixed cryoglobulinemia and cryoglobulinemia-associated disorders.

In this paper we prove a Wiener-type characterization of boundary regularity, in the spirit of a classical result by Landis, for a class of evolutive Hormander operators. We actually show the validity of our criterion for a larger class of degenerate-parabolic operators with a fundamental solution satisfying suitable two-sided Gaussian bounds. Our condition is expressed in terms of a series of balayages or, (as it turns out to be) equivalently, Riesz-potentials.