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Dual pairs of interior and exterior Hardy spaces associated to a simple closed Lipschitz planar curve are considered, leading to a M\"obius invariant function bounding the norm of the Cauchy transform $\bf{C}$ from below. This function is shown to satisfy strong rigidity properties and is closely connected via the Berezin transform to the square of the Kerzman-Stein operator. Explicit example calculations are presented. For ellipses, a new asymptotically sharp lower bound on the norm of $\bf{C}$ is produced., Comment: 28 pages, 3 figures

We present multi-wavelength temporal and spectral characteristics of a magnetic cataclysmic variable (MCV) Swift J0503.7-2819, using far ultraviolet (FUV) and X-ray data from AstroSat, supplemented with optical data from the Southern African Large Telescope and X-ray data from the XMM-Newton and Swift observatories. The X-ray modulations at 4897.6657 s and 3932.0355 s are interpreted as the orbital ($P_{\Omega}$) and spin ($P_{\omega}$) period, respectively, and are consistent with prior reports. With a spin-orbit period ratio of 0.8 and $P_{\Omega}$ falling below the period gap (2-3 hrs) of CVs, Swift J0503.7-2819 would be the newest addition to the growing population of nearly synchronous MCVs, which we call EX Hya-like systems. Hard X-ray luminosity of $<$ $2.5\times10^{32} erg s^{-1}$, as measured with the Swift Burst Alert Telescope, identifies it to be a low-luminosity intermediate polar, similar to other EX Hya-like systems. The phenomenology of the light curves and the spectral characteristics rule out a purely disc-fed/stream-fed model and instead reveal the presence of complex accretion structures around the white dwarf. We propose a ring-like accretion flow, akin to EX Hya, using period ratio, stability arguments, and observational features. An attempt is made to differentiate between the asynchronous polar/nearly-synchronous intermediate polar nature of Swift J0503.7-2819. Further, we note that with the advent of sensitive surveys, a growing population of MCVs that exhibit characteristics of both polars and intermediate polars is beginning to be identified, likely forming a genealogical link between the two conventional classes of MCVs., Comment: Accepted for publication in MNRAS, 17 Pages, 13 Figures

AE Aqr was until recently the only known magnetic cataclysmic variable (MCV) containing a rapidly spinning (33.08 s) white dwarf (WD). Its radio emission is believed to be a superposition of synchrotron emitting plasmoids, because it has a positive spectral index spanning three orders of magnitude (~2-2000 GHz) and is unpolarized. Both characteristics are unusual for MCVs. Recently, Thorstensen has suggested that the cataclysmic variable LAMOST J024048.51+195226.9 (henceforth, J0240+19) is a twin of AE Aqr based on its optical spectra. Optical photometry shows the star to be a high-inclination, eclipsing binary with a spin period of 24.93 s, making it the fastest spinning WD. This paper presents three hours of Very Large Array radio observations of J0240+19. These observations show that the persistent radio emission from J0240+19 is dissimilar to that of AE Aqr in that it shows high circular polarization and a negative spectral index. The emission is most similar to the nova-like CV V603 Aql. We argue that the radio emission is caused by a superposition of plasmoids emitting plasma radiation or electron cyclotron maser emission from the lower corona of the donor star and not from the magnetosphere near the WD, because the latter site is expected to be modulated at the orbital period of the binary and to show eclipses of which there is no evidence. The radio source J0240+19, although weak (<1 mJy), is a persistent source in a high-inclination eclipsing binary, making it a good laboratory for studying radio emission from CVs., Comment: 8 pages, 1 table, 3 figures, minor corrections

The Leray transform $\bf{L}$ is studied on a family $M_\gamma$ of unbounded hypersurfaces in two complex dimensions. For a large class of measures, we obtain necessary and sufficient conditions for the $L^2$-boundedness of $\bf{L}$, along with an exact spectral description of $\bf{L}^*\bf{L}$. This yields both the norm and high-frequency norm of $\bf{L}$, the latter giving an affirmative answer to an unbounded analogue of an open conjecture relating the essential norm of $\bf{L}$ to a projective invariant on a bounded hypersurface. $\bf{L}$ is also shown to play a central role in bridging the function theoretic and projective geometric notions of duality. Our work leads to the construction of projectively invariant Hardy spaces on the $M_\gamma$, along with the realization of their duals as invariant Hardy spaces on the dual hypersurfaces., Comment: 54 pages

A smooth, strongly $\mathbb{C}$-convex, real hypersurface $S$ in $\mathbb{CP}^n$ admits a projective dual CR structure in addition to the standard CR structure. Given a smooth function $u$ on $S$, we provide characterizations for when $u$ can be decomposed as a sum of a CR function and a dual CR function. Following work of Lee on pluriharmonic boundary values, we provide a characterization using differential forms. We further provide a characterization using tangential vector fields in the style of Audibert and Bedford., Comment: 19 pages. To appear in Pure and Applied Mathematics Quarterly