Answer: The answer to your question depends in part on what age/size plants you have planted and how well they are cared for. Astilbes in general need rich soil and ample moisture to grow and bloom their best. If the plants are fairly large and you keep them well watered, then I would expect them to bloom for you but not until later in the summer.
The Hostas will appreciate the same conditions but will grow fine in average soil, too. Most hostas bloom in mid to late summer. Their first year in the ground they may bloom later than normal, or not at all.
The carpet of snow, I believe, must be Alyssum. This plant is an annual and really grows best in full sun but will tolerate part shade. It should definitely bloom this summer but since it is an annual it will not come back next year.
There is another plant, a perennial, called snow in summer (Cerastium tomentosum) which grows best in full sun all day long and well drained soil. It has white flowers and silvery gray foliage. If this is what you have, it may not do so well in a shady area. It would need at least six hours of sun including the hour of noontime (this would actually be considered full sun), or preferably direct sun all day long, or possibly full sun all afternoon since that is very intense sun. It would probably bloom this year if it is getting enough sun.
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